From 3ac127cc8559fe6ccfac36bce72781a2d67f6ade Mon Sep 17 00:00:00 2001 From: Henry Zou Date: Mon, 29 Jun 2026 13:22:23 -0400 Subject: [PATCH 01/14] Add "Automated spacetime Pauli checks with qiskit-paulice" tutorial Add a new Error detection tutorial that uses the qiskit-paulice package to automatically find and insert spacetime Pauli checks into Clifford circuits, then boosts sampled-distribution fidelity by postselecting on check syndromes. It conforms to the tutorial template (Learning outcomes, Prerequisites, Background, Requirements, Setup, small-scale simulator example with Steps 1-4, large-scale hardware example, Next steps, References) and is framed as an extension of the existing ghz-spacetime-codes tutorial, with bidirectional cross-links between the two. Changes: - docs/tutorials/automated-spacetime-pauli-checks.ipynb: new tutorial - docs/tutorials/_toc.json, docs/tutorials/index.mdx: register under "Error detection" (marked isNew) - qiskit_bot.yaml: add owner - scripts/config/notebook-testing.toml: exclude from notebook test runs - docs/tutorials/ghz-spacetime-codes.ipynb: add forward cross-link in Next steps - public/docs/images/tutorials/automated-spacetime-pauli-checks/: extracted plot outputs Note: this run was executed on ibm_marrakesh (Heron r2) and the postselection results are weaker than expected. The qubit layout is hand-picked for ibm_boston (Heron r3) connectivity, so the tutorial needs to be rerun on ibm_boston (or another suitable backend) to investigate the poor results and finalize the outputs, commentary, and usage estimate. --- docs/tutorials/_toc.json | 6 + .../automated-spacetime-pauli-checks.ipynb | 1113 +++++++++++++++++ docs/tutorials/ghz-spacetime-codes.ipynb | 1 + docs/tutorials/index.mdx | 2 + .../hardware-retrieve-1.avif | Bin 0 -> 8560 bytes .../extracted-outputs/step1-code-0.avif | Bin 0 -> 9337 bytes .../extracted-outputs/step2-checks-1.avif | Bin 0 -> 30037 bytes .../extracted-outputs/step2-layout-1.avif | Bin 0 -> 10517 bytes .../extracted-outputs/step2-transpile-0.avif | Bin 0 -> 11621 bytes .../extracted-outputs/step4-fidelity-1.avif | Bin 0 -> 7034 bytes .../extracted-outputs/step4-fidelity-2.avif | Bin 0 -> 7786 bytes .../extracted-outputs/step4-gamma-0.avif | Bin 0 -> 8720 bytes .../extracted-outputs/step4-psr-0.avif | Bin 0 -> 6545 bytes qiskit_bot.yaml | 2 + scripts/config/notebook-testing.toml | 1 + 15 files changed, 1125 insertions(+) create mode 100644 docs/tutorials/automated-spacetime-pauli-checks.ipynb create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/hardware-retrieve-1.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step1-code-0.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step2-checks-1.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step2-layout-1.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step2-transpile-0.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step4-fidelity-1.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step4-fidelity-2.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step4-gamma-0.avif create mode 100644 public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/step4-psr-0.avif diff --git a/docs/tutorials/_toc.json b/docs/tutorials/_toc.json index 6c462de72187..84d065c1085b 100644 --- a/docs/tutorials/_toc.json +++ b/docs/tutorials/_toc.json @@ -243,6 +243,12 @@ { "title": "Low-overhead error detection with spacetime codes", "url": "/docs/tutorials/ghz-spacetime-codes" + }, + { + "title": "Automated spacetime Pauli checks with qiskit-paulice", + "url": "/docs/tutorials/automated-spacetime-pauli-checks", + "isNew": true, + "isNewDate": "2026-06-29" } ] } diff --git a/docs/tutorials/automated-spacetime-pauli-checks.ipynb b/docs/tutorials/automated-spacetime-pauli-checks.ipynb new file mode 100644 index 000000000000..5e67d537a19e --- /dev/null +++ b/docs/tutorials/automated-spacetime-pauli-checks.ipynb @@ -0,0 +1,1113 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "frontmatter", + "metadata": {}, + "source": [ + "---\n", + "title: Automated spacetime Pauli checks with qiskit-paulice\n", + "description: Automatically insert spacetime Pauli checks with qiskit-paulice and boost sampled Clifford-circuit fidelity by postselecting on check syndromes.\n", + "---\n", + "\n", + "# Automated spacetime Pauli checks with qiskit-paulice\n", + "\n", + "*Usage estimate: 1 minute on a Heron r3 processor (NOTE: This is an estimate only. Your runtime may vary.)*" + ] + }, + { + "cell_type": "markdown", + "id": "learning-outcomes", + "metadata": {}, + "source": [ + "## Learning outcomes\n", + "\n", + "After completing this tutorial, you can expect to understand the following information:\n", + "\n", + "- How spacetime Pauli checks detect logical errors in Clifford circuits, and how postselecting on their syndromes boosts the fidelity of a sampled distribution.\n", + "- How to use the `qiskit-paulice` package to find and insert hardware-efficient checks automatically with `get_check_qubits`, `NoiseModel`, and `add_pauli_checks`.\n", + "- How to estimate the fidelity of a stabilizer state by sampling its stabilizers and postselecting on check syndromes.\n", + "- How to run the full error-detection workflow on IBM Quantum® hardware and compare noisy and postselected fidelities." + ] + }, + { + "cell_type": "markdown", + "id": "prerequisites", + "metadata": {}, + "source": [ + "## Prerequisites\n", + "\n", + "It is recommended that you familiarize yourself with these topics:\n", + "\n", + "- The companion tutorial [Low-overhead error detection with spacetime codes](/docs/tutorials/ghz-spacetime-codes), which introduces coherent Pauli checks and builds a single check by hand. This tutorial picks up where that one leaves off and automates the same idea with `qiskit-paulice`.\n", + "- [Hardware fundamentals](/learning/courses/utility-scale-quantum-computing/hardware) for utility-scale quantum computing.\n", + "- The Clifford and stabilizer formalism, including how a stabilizer group describes a pure stabilizer state." + ] + }, + { + "cell_type": "markdown", + "id": "background", + "metadata": {}, + "source": [ + "## Background\n", + "\n", + "[Low-overhead error detection with spacetime codes](https://arxiv.org/abs/2504.15725) [[1]](#references) by Simon Martiel and Ali Javadi-Abhari introduces a method for detecting logical errors in Clifford-dominated circuits that sits between full error correction and lighter-weight error mitigation. The idea builds on coherent Pauli checks (CPC) from [Single-shot error mitigation by coherent Pauli checks](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033193) [[2]](#references) by van den Berg and others. In both approaches, a Clifford \"payload\" circuit is entangled with ancilla qubits to check certain invariants. Measuring the ancillas produces a syndrome that reports whether an error was detected during execution. Keeping only the samples with no detected error improves the fidelity of the sampled distribution, at the cost of a reduced postselection rate.\n", + "\n", + "The key difference between coherent Pauli checks and spacetime checks is the operators they measure. Coherent Pauli checks measure time-localized, high-weight operators. On qubit topologies with limited connectivity, such as heavy hex, those checks need many SWAP gates and often make the circuit too deep to run in practice. Implementing the checks as spacetime codes instead distributes each check across the payload circuit in space and time. This yields a hardware-efficient encoding that stays effective at detecting logical errors while keeping the qubit and depth overhead low.\n", + "\n", + "### What the qiskit-paulice package does\n", + "\n", + "The `qiskit-paulice` package automates the construction of these checks so that you do not have to build them by hand. Its primary role is to find and insert valid spacetime Pauli checks at the locations in a circuit that maximize error detection while minimizing qubit overhead. A check is *valid* when its operators leave the logical action of the payload circuit unchanged, *low-weight* when it uses few entangling gates, and *effective* when it detects a large fraction of the error relative to the noise the check itself introduces. The package scores candidate checks against a noise model and commits the best ones to the circuit. This tutorial uses three pieces of the API:\n", + "\n", + "- `get_check_qubits` inspects a backend coupling map and returns target and ancilla qubit pairs. A check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", + "- `NoiseModel.from_backend` builds a rough noise model from backend benchmark data. The model scores candidate checks, so an exact, learned noise model is not required. For a learned Pauli-Lindblad model, see `NoiseModel.from_pauli_lindblad_maps`.\n", + "- `add_pauli_checks` finds and inserts checks into a circuit. It returns a sequence of `CheckedCircuit` objects with an increasing number of checks, and each object provides a `get_postselection_method` that maps a measured bitstring to a syndrome vector. The `cost` argument selects the function that scores a check (`gamma`, the sampling overhead of the postselected inverse noise channel, or `LER`, the logical error rate). The `method` argument selects the search strategy (`windowed`, `genetic`, or `windowed_genetic`). This tutorial uses `cost=\"gamma\"` and `method=\"windowed\"`, which together give deterministic, reproducible check selection.\n", + "\n", + "### Estimating fidelity from stabilizer sampling\n", + "\n", + "To measure how well the error detection works, you can estimate the fidelity of the stabilizer state $|\\psi\\rangle = U|0\\rangle^{\\otimes n}$ that the circuit ideally prepares against the noisy state $\\rho$ that the hardware actually outputs. The projector onto a pure stabilizer state $|\\psi\\rangle$ equals the uniform average over the $2^n$ elements of its stabilizer group $\\mathcal{S}$:\n", + "\n", + "$$|\\psi\\rangle\\langle\\psi| = \\frac{1}{2^n}\\sum_{G \\in \\mathcal{S}} G.$$\n", + "\n", + "Substituting this into the fidelity gives the fidelity of $\\rho$ as the average expectation value of every stabilizer $G \\in \\mathcal{S}$ with respect to $\\rho$:\n", + "\n", + "$$F = \\mathrm{Tr}(\\rho|\\psi\\rangle\\langle\\psi|) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\mathrm{Tr}(\\rho G) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\langle G \\rangle_\\rho.$$\n", + "\n", + "For larger problems, enumerating all $2^n$ stabilizers is infeasible, so you can estimate the fidelity from a random sample. Drawing $M$ stabilizers $G_1, \\ldots, G_M$ uniformly at random from $\\mathcal{S}$ gives an unbiased estimator:\n", + "\n", + "$$\\hat F_M = \\frac{1}{M} \\sum_{i=1}^M \\langle G_i \\rangle_\\rho.$$\n", + "\n", + "Because a Clifford circuit prepares a stabilizer state, you can estimate its fidelity directly from sampled expectation values of its stabilizers. This tutorial first walks through the workflow on a simulator with a small circuit, then runs the same workflow on hardware with a deeper circuit. As circuits include more non-Clifford operations the number of valid checks shrinks quickly, so the method works best for Clifford-dominated circuits." + ] + }, + { + "cell_type": "markdown", + "id": "requirements", + "metadata": {}, + "source": [ + "## Requirements\n", + "\n", + "Before starting this tutorial, be sure you have the following installed:\n", + "\n", + "- Qiskit SDK v2.0 or later, with [visualization](/docs/api/qiskit/visualization) support\n", + "- Qiskit Runtime v0.40 or later (`pip install qiskit-ibm-runtime`)\n", + "- Qiskit Aer v0.17 or later (`pip install qiskit-aer`)\n", + "- Qiskit Paulice (`pip install qiskit-paulice`)\n", + "- tqdm (`pip install tqdm`)" + ] + }, + { + "cell_type": "markdown", + "id": "setup", + "metadata": {}, + "source": [ + "## Setup\n", + "\n", + "Import the required libraries and define the helper functions that are not available as imports. The `random_clifford_circuit` function builds a brickwork random Clifford payload, `append_basis_rotation` rotates a circuit so that a stabilizer is measured in the computational basis, `expectation` computes a stabilizer expectation value from sampled counts, and `cum_mean_sem` tracks the running fidelity estimate." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "setup-imports", + "metadata": {}, + "outputs": [], + "source": [ + "# Standard library imports\n", + "import random\n", + "import time\n", + "\n", + "# External libraries\n", + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "from tqdm import tqdm\n", + "\n", + "# Qiskit\n", + "from qiskit import QuantumCircuit\n", + "from qiskit.quantum_info import Clifford, Pauli, PauliList\n", + "from qiskit.result import sampled_expectation_value\n", + "from qiskit.transpiler import generate_preset_pass_manager\n", + "from qiskit.visualization import plot_coupling_map\n", + "\n", + "# Qiskit Aer\n", + "from qiskit_aer import AerSimulator\n", + "from qiskit_aer.noise import NoiseModel as AerNoiseModel\n", + "from qiskit_aer.noise import ReadoutError, depolarizing_error\n", + "\n", + "# Qiskit IBM Runtime\n", + "from qiskit_ibm_runtime import QiskitRuntimeService\n", + "from qiskit_ibm_runtime import SamplerV2 as Sampler\n", + "\n", + "# Qiskit Paulice\n", + "from qiskit_paulice import add_pauli_checks\n", + "from qiskit_paulice.layout import get_check_qubits\n", + "from qiskit_paulice.noise_models import NoiseModel" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "setup-helpers", + "metadata": {}, + "outputs": [], + "source": [ + "def random_clifford_circuit(\n", + " num_qubits: int, depth: int, rng: np.random.Generator\n", + ") -> QuantumCircuit:\n", + " \"\"\"Brickwork random Clifford on `num_qubits`, with `depth` CZ layers.\"\"\"\n", + " qc = QuantumCircuit(num_qubits)\n", + " qc.h(range(num_qubits))\n", + " for d in range(depth):\n", + " for i in range(d % 2, num_qubits - 1, 2):\n", + " qc.cz(i, i + 1)\n", + " for q in range(num_qubits):\n", + " if rng.integers(0, 2):\n", + " qc.sx(q)\n", + " if rng.integers(0, 2):\n", + " qc.s(q)\n", + " if rng.integers(0, 2):\n", + " qc.sx(q)\n", + " return qc\n", + "\n", + "\n", + "def append_basis_rotation(\n", + " circuit: QuantumCircuit, pauli: Pauli\n", + ") -> QuantumCircuit:\n", + " \"\"\"Strip measurements, append basis rotations for `pauli`, and re-measure.\"\"\"\n", + " out = circuit.remove_final_measurements(inplace=False)\n", + " for q in range(pauli.num_qubits):\n", + " if pauli.x[q]:\n", + " if pauli.z[q]:\n", + " out.sdg(q)\n", + " out.h(q)\n", + " out.measure_all()\n", + " return out\n", + "\n", + "\n", + "def expectation(counts: dict, pauli: Pauli) -> float:\n", + " \"\"\"Expectation value of `pauli` from counts measured in the Z basis.\n", + "\n", + " Pads with identity on any qubits beyond the support of `pauli`, such as the\n", + " check ancillas that appear in the postselected counts.\n", + " \"\"\"\n", + " if not counts:\n", + " return float(\"nan\")\n", + " n = pauli.num_qubits\n", + " sign = -1 if int(pauli.phase) % 4 == 2 else 1\n", + " total = len(next(iter(counts)))\n", + " label = \"\".join(\n", + " \"Z\" if q < n and (pauli.x[q] or pauli.z[q]) else \"I\"\n", + " for q in range(total - 1, -1, -1)\n", + " )\n", + " return sign * sampled_expectation_value(counts, label)\n", + "\n", + "\n", + "def cum_mean_sem(values: np.ndarray):\n", + " \"\"\"Cumulative mean and standard error of the mean, ignoring NaNs.\"\"\"\n", + " valid = ~np.isnan(values)\n", + " total = np.cumsum(np.where(valid, values, 0.0))\n", + " total_sq = np.cumsum(np.where(valid, values**2, 0.0))\n", + " count = np.maximum(np.cumsum(valid).astype(float), 1)\n", + " mean = total / count\n", + " sem = np.sqrt(np.maximum(total_sq / count - mean**2, 0) / count)\n", + " return np.where(np.cumsum(valid) > 0, mean, np.nan), sem" + ] + }, + { + "cell_type": "markdown", + "id": "small-scale-header", + "metadata": {}, + "source": [ + "## Small-scale simulator example\n", + "\n", + "This section walks through the full workflow on a noisy simulator. It uses backend benchmark data to choose a qubit layout and a noise model, finds checks automatically, and uses postselection on the sampled distribution to show the fidelity improvement." + ] + }, + { + "cell_type": "markdown", + "id": "step1-header", + "metadata": {}, + "source": [ + "### Step 1: Map classical inputs to a quantum problem\n", + "\n", + "The payload circuit is a shallow one-dimensional brickwork random Clifford circuit. Because the circuit is Clifford, it prepares a stabilizer state whose fidelity you can estimate directly from sampled stabilizer expectation values. Start with a shallow circuit so that the checks are easy to visualize in the next step." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "step1-code", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "num_qubits = 12\n", + "depth = 4\n", + "seed = 1764\n", + "rng = np.random.default_rng(seed)\n", + "np.random.seed(seed)\n", + "\n", + "circuit = random_clifford_circuit(num_qubits, depth, rng)\n", + "circuit.measure_all()\n", + "circuit.draw(\"mpl\", fold=-1, scale=0.6)" + ] + }, + { + "cell_type": "markdown", + "id": "step2-header", + "metadata": {}, + "source": [ + "### Step 2: Optimize problem for quantum hardware execution\n", + "\n", + "Mapping the circuit to hardware sets the physical qubit layout, the noise model that scores candidate checks, and the checks themselves.\n", + "\n", + "First, choose a one-dimensional qubit layout on the Heron r3 QPU `ibm_boston`. The layout zig-zags diagonally through the coupling graph to maximize the number of available ancilla qubits. The `get_check_qubits` function returns target and ancilla pairs, where a check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", + "\n", + "In the coupling graph that follows, the green qubits are payload qubits and the orange qubits are the ancillas that implement the checks. Qubits with an adjacent ancilla are used as target qubits for checks." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "id": "step2-layout", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Target qubits: [68, 69, 89, 91, 111, 113, 133]\n", + "Ancilla qubits: [67, 70, 88, 92, 110, 114, 132]\n" + ] + }, + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "service = QiskitRuntimeService()\n", + "backend = service.backend(\"ibm_marrakesh\")\n", + "\n", + "# Choose a layout and pair each available target qubit with a neighboring ancilla\n", + "layout = [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133]\n", + "target_qubits, ancilla_qubits = get_check_qubits(backend, layout)\n", + "num_checks = len(target_qubits)\n", + "\n", + "print(f\"Target qubits: {target_qubits}\")\n", + "print(f\"Ancilla qubits: {ancilla_qubits}\")\n", + "plot_coupling_map(\n", + " num_qubits=backend.num_qubits,\n", + " qubit_coordinates=getattr(\n", + " backend.configuration(), \"qubit_coordinates\", None\n", + " ),\n", + " coupling_map=backend.configuration().coupling_map,\n", + " figsize=(12, 12),\n", + " qubit_color=[\n", + " \"#4CAF50\"\n", + " if i in set(layout)\n", + " else \"#FF9800\"\n", + " if i in set(ancilla_qubits)\n", + " else \"#DDDDDD\"\n", + " for i in backend.coupling_map.graph.node_indices()\n", + " ],\n", + " qubit_size=220,\n", + " line_width=2,\n", + " font_size=90,\n", + ")" + ] + }, + { + "cell_type": "markdown", + "id": "step2-transpile-md", + "metadata": {}, + "source": [ + "With the backend and layout chosen, transpile the payload into an instruction set architecture (ISA) circuit. Setting the layout and translating the gates into the native gate set of `ibm_boston` is enough here." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "id": "step2-transpile", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pm = generate_preset_pass_manager(\n", + " optimization_level=0, backend=backend, initial_layout=layout\n", + ")\n", + "circuit_isa = pm.run(circuit)\n", + "circuit_isa.draw(\"mpl\", fold=-1, scale=0.6)" + ] + }, + { + "cell_type": "markdown", + "id": "step2-noise-md", + "metadata": {}, + "source": [ + "Next, model how the gate and readout noise on the backend affects execution. The noise model determines where in the circuit a check captures the most error. A more accurate model improves the detection, but it is usually not necessary to learn one by sampling the QPU. The model that follows infers a uniform depolarizing channel for gate and readout noise from `qiskit-ibm-runtime` benchmark data." + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "id": "step2-noise", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "NoiseModel(gate_noise=0.19074977552838765, readout_noise=0.09235636393229167, idling_noise=None)\n" + ] + } + ], + "source": [ + "noise_model = NoiseModel.from_backend(\n", + " backend, layout, uniform_gate_noise=True\n", + ")\n", + "print(noise_model)" + ] + }, + { + "cell_type": "markdown", + "id": "step2-checks-md", + "metadata": {}, + "source": [ + "Now add checks to the circuit. The `add_pauli_checks` function takes the Clifford payload, the list of target qubits, and the noise model. The `ancilla_qubits` argument tells the function which physical ancilla to pair with each target. Checks are added in the order the target qubits appear, so the final layout of the checked circuit is `layout + ancilla_qubits`. To run an output circuit with fewer (`i`) checks, the final layout is `layout + ancilla_qubits[:i]`.\n", + "\n", + "The output of `add_pauli_checks` is a sequence of circuits with an increasing number of checks, from no checks up to one check on every target qubit. The visualization confirms that the checks use the specified target and ancilla pairs. For details on finding good checks, see Sections II to IV of the supplementary information in reference [[1]](#references)." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "id": "step2-checks", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Physical layout of payload and ancillas: [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133, 67, 70, 88, 92, 110, 114, 132]\n", + "Checked circuit:\n" + ] + }, + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "checked = add_pauli_checks(\n", + " circuit_isa,\n", + " target_qubits,\n", + " noise_model,\n", + " ancilla_qubits=ancilla_qubits,\n", + " cost=\"gamma\",\n", + " method=\"windowed\",\n", + " seed=seed,\n", + ")\n", + "\n", + "print(f\"Physical layout of payload and ancillas: {layout + ancilla_qubits}\")\n", + "print(\"Checked circuit:\")\n", + "checked[-1].circuit.draw(\"mpl\", fold=-1, idle_wires=False)" + ] + }, + { + "cell_type": "markdown", + "id": "step3-header", + "metadata": {}, + "source": [ + "### Step 3: Execute using Qiskit primitives\n", + "\n", + "To make the effect of gate noise visible, increase the depth of the payload and sample a subset of its stabilizers. Each stabilizer is generally not qubit-wise commuting with the others, so a single set of checks is not valid for two different stabilizers. Rather than group stabilizers into commuting sets, find a good set of checks for each stabilizer independently. Sampling stabilizers uniformly at random gives an unbiased fidelity estimate.\n", + "\n", + "Build the deeper circuit and draw a random sample of its stabilizers." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "id": "step3-stabilizers", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {-XYXZIZXYIIXZ, -XZYYIIXZXYYY, ...}\n" + ] + } + ], + "source": [ + "depth = 24\n", + "num_stabilizers = 20\n", + "num_shots = 1_000\n", + "\n", + "circuit = random_clifford_circuit(num_qubits, depth, rng)\n", + "\n", + "# Build the full stabilizer group, then sample from it uniformly at random\n", + "circ_no_meas = circuit.remove_final_measurements(inplace=False)\n", + "stabilizer_group = PauliList([Pauli(\"I\" * num_qubits)])\n", + "for generator in (\n", + " Pauli(label) for label in Clifford(circ_no_meas).to_labels(mode=\"S\")\n", + "):\n", + " stabilizer_group = stabilizer_group + stabilizer_group.compose(generator)\n", + "\n", + "keep = np.where(\n", + " stabilizer_group.x.any(axis=1) | stabilizer_group.z.any(axis=1)\n", + ")[0]\n", + "chosen = np.random.default_rng(seed).choice(\n", + " keep, size=min(num_stabilizers, len(keep)), replace=False\n", + ")\n", + "stabilizers = [stabilizer_group[int(i)] for i in chosen]\n", + "\n", + "two_qubit_depth = circuit.depth(lambda x: x.operation.num_qubits == 2)\n", + "print(\n", + " f\"Sampled {len(stabilizers)} stabilizers of a {circuit.num_qubits}-qubit \"\n", + " f\"circuit with two-qubit depth {two_qubit_depth}: \"\n", + " f\"{{{stabilizers[0]}, {stabilizers[1]}, ...}}\"\n", + ")" + ] + }, + { + "cell_type": "markdown", + "id": "step3-findchecks-md", + "metadata": {}, + "source": [ + "For each sampled stabilizer, rotate the circuit so that the stabilizer is measured in the computational basis, transpile it onto the backend, and find a good set of checks. The target and ancilla pairs are shuffled together for each stabilizer so that each target keeps its ancilla. Remember that checks are committed sequentially in the order the target qubits are given, and a committed check is not changed as more checks are added." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "step3-findchecks", + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 20/20 [00:15<00:00, 1.32it/s]" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Added 7 checks to 20 circuits in 15s.\n", + "On average, two-qubit depth increased from 24 to 32 when adding 7 checks.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "\n" + ] + } + ], + "source": [ + "noisy_circuits = []\n", + "checked_circuits = []\n", + "depths_2q = []\n", + "t0 = time.time()\n", + "for i, pauli in enumerate(tqdm(stabilizers)):\n", + " noisy_circuits.append(pm.run(append_basis_rotation(circuit, pauli)))\n", + " # Shuffle target and ancilla pairs together so each target keeps its ancilla\n", + " targets, ancillas = zip(\n", + " *random.sample(\n", + " list(zip(target_qubits, ancilla_qubits, strict=True)),\n", + " k=len(target_qubits),\n", + " ),\n", + " strict=True,\n", + " )\n", + " checked_circuits.append(\n", + " add_pauli_checks(\n", + " noisy_circuits[-1],\n", + " list(targets),\n", + " noise_model,\n", + " ancilla_qubits=list(ancillas),\n", + " cost=\"gamma\",\n", + " method=\"windowed\",\n", + " seed=seed + 1 + i,\n", + " )\n", + " )\n", + " depths_2q.append(\n", + " checked_circuits[-1][-1].circuit.depth(lambda x: len(x.qubits) == 2)\n", + " )\n", + "\n", + "print(\n", + " f\"Added {num_checks} checks to {len(stabilizers)} circuits \"\n", + " f\"in {(time.time() - t0):.0f}s.\"\n", + ")\n", + "print(\n", + " f\"On average, two-qubit depth increased from \"\n", + " f\"{circuit.depth(lambda x: len(x.qubits) == 2)} to {int(np.mean(depths_2q))} \"\n", + " f\"when adding {num_checks} checks.\"\n", + ")" + ] + }, + { + "cell_type": "markdown", + "id": "step3-sample-md", + "metadata": {}, + "source": [ + "Sample the bare payload and the checked circuits with Qiskit Aer. The simulator uses the same depolarizing model that scored the checks, so the noise that the checks target is the noise that the simulator applies." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "step3-sample", + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 20/20 [00:18<00:00, 1.11it/s]\n" + ] + } + ], + "source": [ + "aer_nm = AerNoiseModel()\n", + "aer_nm.add_all_qubit_quantum_error(\n", + " depolarizing_error(noise_model.gate_noise, 2), [\"cz\"]\n", + ")\n", + "p = noise_model.readout_noise\n", + "aer_nm.add_all_qubit_readout_error(ReadoutError([[1 - p, p], [p, 1 - p]]))\n", + "noisy_sim = AerSimulator(method=\"stabilizer\", noise_model=aer_nm)\n", + "\n", + "counts = []\n", + "for i, checked_circ_result in enumerate(tqdm(checked_circuits)):\n", + " noisy_counts = (\n", + " noisy_sim.run(\n", + " noisy_circuits[i], shots=num_shots, seed_simulator=seed * i + 1\n", + " )\n", + " .result()\n", + " .get_counts()\n", + " )\n", + " checked_counts_per_variant = []\n", + " for k, ck in enumerate(checked_circ_result):\n", + " variant_counts = (\n", + " noisy_sim.run(\n", + " ck.circuit, shots=num_shots, seed_simulator=seed * i + 2 + k\n", + " )\n", + " .result()\n", + " .get_counts()\n", + " )\n", + " checked_counts_per_variant.append(variant_counts)\n", + " counts.append((noisy_counts, checked_counts_per_variant))" + ] + }, + { + "cell_type": "markdown", + "id": "step4-header", + "metadata": {}, + "source": [ + "### Step 4: Post-process and return result in desired classical format\n", + "\n", + "Each check uses entangling gates between one ancilla and one target. The ancilla starts in $|0\\rangle$, so $Z_\\text{anc}$ stabilizes its input. Propagating $Z_\\text{anc}$ forward through the checked circuit yields a Pauli operator on the output whose non-identity terms define the check's support. A check passes when the bits in its support have even parity. A sample is kept only when every check passes.\n", + "\n", + "The `get_postselection_method` of each `CheckedCircuit` returns a function that maps a measured bitstring to a syndrome vector. Keep the samples whose syndrome is zero for every check, and discard the rest. The chart that follows shows that adding more checks lowers the postselection rate. A lower postselection rate needs more shots to reach a target accuracy, so there is a tradeoff between detection capability and sampling cost. The rate appears to converge, which indicates that additional checks contribute less detection capability." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "id": "step4-psr", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "rate_per_variant = []\n", + "kept_per_stab = []\n", + "for i, (_, checked_counts_per_variant) in enumerate(counts):\n", + " rates = []\n", + " kept_at_num_checks = None\n", + " for k, variant_counts in enumerate(checked_counts_per_variant):\n", + " ps_fn = checked_circuits[i][k].get_postselection_method()\n", + " kept = {\n", + " bs: n for bs, n in variant_counts.items() if not ps_fn(bs).any()\n", + " }\n", + " rates.append(sum(kept.values()) / num_shots)\n", + " if k == num_checks:\n", + " kept_at_num_checks = kept\n", + " rate_per_variant.append(rates)\n", + " kept_per_stab.append(kept_at_num_checks)\n", + "\n", + "max_len = max(len(s) for s in rate_per_variant)\n", + "rates_arr = np.full((len(rate_per_variant), max_len), np.nan)\n", + "for i, s in enumerate(rate_per_variant):\n", + " rates_arr[i, : len(s)] = s\n", + "ks = np.arange(max_len)\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.plot(ks, rates_arr.T, color=\"#ff8c00\", alpha=0.15, linewidth=1)\n", + "ax.plot(\n", + " ks,\n", + " np.nanmedian(rates_arr, axis=0),\n", + " color=\"black\",\n", + " linewidth=1,\n", + " linestyle=\"--\",\n", + " label=\"median\",\n", + ")\n", + "ax.set_xlabel(\"Checks committed\")\n", + "ax.set_ylabel(\"Postselection rate\")\n", + "ax.set_ylim((0, 1.05))\n", + "ax.set_title(\n", + " f\"Per-stabilizer postselection rate ({len(rates_arr)} stabilizers)\"\n", + ")\n", + "ax.legend()\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "step4-fidelity-md", + "metadata": {}, + "source": [ + "Now compare the fidelity of the bare noisy state with the postselected state. Postselecting only the samples with no detected error raises the expectation value of every stabilizer, and therefore the estimated fidelity. The postselected values use fewer samples than the raw values, yet the expectation values are more accurate and the sampled variance is lower." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "id": "step4-fidelity", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ideal fidelity: 1.0\n", + "noisy fidelity: -0.0058\n", + "postselected fidelity: -0.1257\n", + "mean postselection rate: 0.007\n" + ] + }, + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "results = []\n", + "for i, ((noisy_counts, _), kept) in enumerate(\n", + " zip(counts, kept_per_stab, strict=True)\n", + "):\n", + " results.append(\n", + " (\n", + " expectation(noisy_counts, stabilizers[i]),\n", + " expectation(kept, stabilizers[i]),\n", + " sum(kept.values()) / num_shots,\n", + " )\n", + " )\n", + "\n", + "fidelity_noisy = float(np.nanmean([r[0] for r in results]))\n", + "fidelity_postsel = float(np.nanmean([r[1] for r in results]))\n", + "psr = float(np.mean([r[2] for r in results]))\n", + "print(\n", + " f\"ideal fidelity: 1.0\\n\"\n", + " f\"noisy fidelity: {fidelity_noisy:.4f}\\n\"\n", + " f\"postselected fidelity: {fidelity_postsel:.4f}\\n\"\n", + " f\"mean postselection rate: {psr:.3f}\"\n", + ")\n", + "\n", + "evs_ideal = np.ones(len(results))\n", + "evs_noisy = np.array([r[0] for r in results])\n", + "evs_post = np.array([r[1] for r in results])\n", + "idx = np.arange(len(results))\n", + "\n", + "\n", + "def strip(ax, ys, color, label):\n", + " m, s = np.nanmean(ys), np.nanstd(ys)\n", + " ax.axhspan(\n", + " m - s, m + s, color=color, alpha=0.15, label=f\"{label} mean and std\"\n", + " )\n", + " ax.axhline(m, color=color, linewidth=1, linestyle=\"--\")\n", + "\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.axhline(np.nanmean(evs_ideal), color=\"black\", linewidth=1.5, label=\"ideal\")\n", + "strip(ax, evs_noisy, \"red\", \"noisy\")\n", + "strip(ax, evs_post, \"green\", \"postselected\")\n", + "ax.scatter(idx, evs_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", + "ax.scatter(\n", + " idx,\n", + " evs_post,\n", + " color=\"green\",\n", + " s=22,\n", + " alpha=0.7,\n", + " label=\"postselected\",\n", + ")\n", + "ax.set_xlabel(\"stabilizer index\")\n", + "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", + "ax.set_ylim((-0.1, 1.1))\n", + "ax.set_title(\"Per-stabilizer expectation values\")\n", + "ax.legend(loc=\"lower left\")\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()\n", + "\n", + "M = np.arange(1, len(results) + 1)\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "for ys, color, label in [\n", + " (evs_ideal, \"black\", \"ideal\"),\n", + " (evs_noisy, \"red\", \"noisy\"),\n", + " (evs_post, \"green\", \"postselected\"),\n", + "]:\n", + " cm, sem = cum_mean_sem(ys)\n", + " ax.plot(M, cm, color=color, linewidth=1.5, label=label)\n", + " ax.fill_between(M, cm - sem, cm + sem, color=color, alpha=0.15)\n", + "ax.set_xlabel(\"number of stabilizers averaged\")\n", + "ax.set_ylabel(\"running fidelity estimate\")\n", + "ax.set_title(\"Fidelity convergence versus number of stabilizers\")\n", + "ax.legend()\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "step4-gamma-md", + "metadata": {}, + "source": [ + "The gamma score reports how much of the modeled noise channel remains undetected by the checks. Plotting the gamma score against the number of committed checks shows how the detection capability improves as each check is added. A gamma curve that converges toward `1.0` indicates that additional checks detect little extra error, which matches the convergence seen in the postselection rate." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "id": "step4-gamma", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "stab_scores = [\n", + " [variant.cost for variant in checked_circ_result]\n", + " for checked_circ_result in checked_circuits\n", + "]\n", + "max_len = max(len(s) for s in stab_scores)\n", + "scores = np.full((len(stab_scores), max_len), np.nan)\n", + "for i, s in enumerate(stab_scores):\n", + " scores[i, : len(s)] = s\n", + "ks = np.arange(max_len)\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.plot(ks, scores.T, color=\"#4682b4\", alpha=0.15, linewidth=1)\n", + "ax.plot(\n", + " ks,\n", + " np.nanmedian(scores, axis=0),\n", + " color=\"black\",\n", + " linewidth=1,\n", + " linestyle=\"--\",\n", + " label=\"median\",\n", + ")\n", + "ax.set_xlabel(\"Checks committed\")\n", + "ax.set_ylabel(\"Gamma\")\n", + "ax.set_yscale(\"log\")\n", + "ax.set_title(f\"Per-stabilizer gamma curves ({len(scores)} stabilizers)\")\n", + "ax.legend()\n", + "ax.grid(True, alpha=0.3, which=\"both\")\n", + "plt.tight_layout()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "hardware-header", + "metadata": {}, + "source": [ + "## Large-scale hardware example\n", + "\n", + "The same workflow runs on hardware with a deeper payload and more stabilizers. This section reuses the backend, layout, target and ancilla pairs, noise model, and pass manager from the simulator example, then submits the sampled circuits to the QPU in a single job. Because the circuit is deeper, gate noise has a larger effect, and the fidelity gain from postselection is more pronounced.\n", + "\n", + "The parameters that follow set the depth, the number of stabilizers, and the number of shots. Increase `hw_num_stabilizers` for a more precise fidelity estimate at the cost of more circuits per job, and adjust the layout to use more qubits if you want a larger payload." + ] + }, + { + "cell_type": "markdown", + "id": "hardware-steps-header", + "metadata": {}, + "source": [ + "### Steps 1-4 (compressed into a single code block)\n", + "\n", + "The first cell builds the deeper payload, samples its stabilizers, finds the fully checked circuit for each one, and submits one Sampler job that contains both the bare and the checked circuits. Each job carries the tag `TUT_ASPC` so that you can find it later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "id": "hardware-submit", + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 50/50 [03:33<00:00, 4.27s/it]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Submitted job d919t1fqq29s738n85cg with 100 circuits\n" + ] + } + ], + "source": [ + "# -------------------------Step 1: build a deeper payload and sample its stabilizers-------------------------\n", + "hw_depth = 72\n", + "hw_num_stabilizers = 50\n", + "hw_num_shots = 2_000\n", + "\n", + "hw_circuit = random_clifford_circuit(num_qubits, hw_depth, rng)\n", + "hw_no_meas = hw_circuit.remove_final_measurements(inplace=False)\n", + "hw_group = PauliList([Pauli(\"I\" * num_qubits)])\n", + "for generator in (\n", + " Pauli(label) for label in Clifford(hw_no_meas).to_labels(mode=\"S\")\n", + "):\n", + " hw_group = hw_group + hw_group.compose(generator)\n", + "keep = np.where(hw_group.x.any(axis=1) | hw_group.z.any(axis=1))[0]\n", + "chosen = np.random.default_rng(seed).choice(\n", + " keep, size=min(hw_num_stabilizers, len(keep)), replace=False\n", + ")\n", + "hw_stabilizers = [hw_group[int(i)] for i in chosen]\n", + "\n", + "# -------------------------Step 2: transpile and add the fully checked circuit per stabilizer-------------------------\n", + "hw_noisy_circuits = []\n", + "hw_checked_circuits = []\n", + "for i, pauli in enumerate(tqdm(hw_stabilizers)):\n", + " bare = pm.run(append_basis_rotation(hw_circuit, pauli))\n", + " hw_noisy_circuits.append(bare)\n", + " variants = add_pauli_checks(\n", + " bare,\n", + " target_qubits,\n", + " noise_model,\n", + " ancilla_qubits=ancilla_qubits,\n", + " cost=\"gamma\",\n", + " method=\"windowed\",\n", + " seed=seed + 1 + i,\n", + " )\n", + " hw_checked_circuits.append(variants[-1]) # keep the fully checked circuit\n", + "\n", + "# -------------------------Step 3: submit one Sampler job with the bare and checked circuits-------------------------\n", + "sampler = Sampler(mode=backend)\n", + "sampler.options.default_shots = hw_num_shots\n", + "sampler.options.environment.job_tags = [\"TUT_ASPC\"]\n", + "\n", + "pubs = hw_noisy_circuits + [cc.circuit for cc in hw_checked_circuits]\n", + "job = sampler.run(pubs)\n", + "print(f\"Submitted job {job.job_id()} with {len(pubs)} circuits\")" + ] + }, + { + "cell_type": "markdown", + "id": "hardware-retrieve-md", + "metadata": {}, + "source": [ + "After the job completes, retrieve the results and postselect the checked counts. Compute the noisy and postselected fidelities as the average stabilizer expectation value over the sampled stabilizers." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "id": "hardware-retrieve", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "noisy fidelity: 0.0000\n", + "postselected fidelity: 0.0125\n", + "mean postselection rate: 0.008\n" + ] + }, + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "# -------------------------Step 4: postselect and compare fidelity-------------------------\n", + "result = job.result()\n", + "n_stab = len(hw_stabilizers)\n", + "\n", + "hw_results = []\n", + "for i in range(n_stab):\n", + " noisy_counts = result[i].join_data().get_counts()\n", + " checked_counts = result[n_stab + i].join_data().get_counts()\n", + " ps_fn = hw_checked_circuits[i].get_postselection_method()\n", + " kept = {bs: c for bs, c in checked_counts.items() if not ps_fn(bs).any()}\n", + " hw_results.append(\n", + " (\n", + " expectation(noisy_counts, hw_stabilizers[i]),\n", + " expectation(kept, hw_stabilizers[i]),\n", + " sum(kept.values()) / sum(checked_counts.values()),\n", + " )\n", + " )\n", + "\n", + "hw_fidelity_noisy = float(np.nanmean([r[0] for r in hw_results]))\n", + "hw_fidelity_postsel = float(np.nanmean([r[1] for r in hw_results]))\n", + "hw_psr = float(np.mean([r[2] for r in hw_results]))\n", + "print(\n", + " f\"noisy fidelity: {hw_fidelity_noisy:.4f}\\n\"\n", + " f\"postselected fidelity: {hw_fidelity_postsel:.4f}\\n\"\n", + " f\"mean postselection rate: {hw_psr:.3f}\"\n", + ")\n", + "\n", + "hw_noisy = np.array([r[0] for r in hw_results])\n", + "hw_post = np.array([r[1] for r in hw_results])\n", + "idx = np.arange(n_stab)\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.axhline(1.0, color=\"black\", linewidth=1.5, label=\"ideal\")\n", + "strip(ax, hw_noisy, \"red\", \"noisy\")\n", + "strip(ax, hw_post, \"green\", \"postselected\")\n", + "ax.scatter(idx, hw_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", + "ax.scatter(\n", + " idx,\n", + " hw_post,\n", + " color=\"green\",\n", + " s=22,\n", + " alpha=0.7,\n", + " label=\"postselected\",\n", + ")\n", + "ax.set_xlabel(\"stabilizer index\")\n", + "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", + "ax.set_ylim((-0.1, 1.1))\n", + "ax.set_title(\"Per-stabilizer expectation values on hardware\")\n", + "ax.legend(loc=\"lower left\")\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "next-steps", + "metadata": {}, + "source": [ + "## Next steps\n", + "\n", + "\n", + "If you found this work interesting, you might be interested in the following material:\n", + "\n", + "- The conceptual companion tutorial [Low-overhead error detection with spacetime codes](/docs/tutorials/ghz-spacetime-codes), which builds a coherent Pauli check by hand and prepares a GHZ state on hardware.\n", + "- The tutorial on [repetition codes](/docs/tutorials/repetition-codes) for an introduction to quantum error correction.\n", + "- The [`qiskit-paulice` package](https://github.com/Qiskit/qiskit-paulice) for the full check-finding API.\n", + "- The paper [Low-overhead error detection with spacetime codes](https://arxiv.org/abs/2504.15725) for the theory behind the checks.\n", + "" + ] + }, + { + "cell_type": "markdown", + "id": "references", + "metadata": {}, + "source": [ + "## References\n", + "\n", + "- [1] Martiel, S., & Javadi-Abhari, A. (2025). Low-overhead error detection with spacetime codes. *arXiv preprint* [arXiv:2504.15725](https://arxiv.org/abs/2504.15725).\n", + "- [2] van den Berg, E., Bravyi, S., Gambetta, J. M., Jurcevic, P., Maslov, D., & Temme, K. (2023). Single-shot error mitigation by coherent Pauli checks. *Physical Review Research*, 5(3), 033193." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index 49382566ecd8..ff1e2e3fc626 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -1859,6 +1859,7 @@ "If you found this work interesting, you might be interested in the following material:\n", "- To further explore technical details of GHZ state preparation, see [[3]](https://arxiv.org/abs/2510.09520). In addition to error detection, authors use readout error mitigation with M3 and TREX and perform temporary uncomputation techniques to prepare high-fidelity GHZ states.\n", "- Tutorial on [repetition codes](/docs/tutorials/repetition-codes).\n", + "- The tutorial [Automated spacetime Pauli checks with qiskit-paulice](/docs/tutorials/automated-spacetime-pauli-checks), which automates check finding with the `qiskit-paulice` package and boosts sampled Clifford-circuit fidelity through postselection.\n", "" ] }, diff --git a/docs/tutorials/index.mdx b/docs/tutorials/index.mdx index 3ad267b2f46c..828605fb95c4 100644 --- a/docs/tutorials/index.mdx +++ b/docs/tutorials/index.mdx @@ -173,6 +173,8 @@ Error detection identifies faulty operations to return noise-free results shot-b * [Low-overhead error detection with spacetime codes](/docs/tutorials/ghz-spacetime-codes) +* [Automated spacetime Pauli checks with qiskit-paulice](/docs/tutorials/automated-spacetime-pauli-checks) + diff --git 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"`@nathanearnestnoble`" - "`@MeltemTolunay`" + "docs/tutorials/automated-spacetime-pauli-checks": + - "`@henryzou50`" "docs/tutorials/spin-chain-vqe": - "`@nathanearnestnoble`" - "`@mrvee-qC-bee`" diff --git a/scripts/config/notebook-testing.toml b/scripts/config/notebook-testing.toml index 001f83bcbed6..074d81e00164 100644 --- a/scripts/config/notebook-testing.toml +++ b/scripts/config/notebook-testing.toml @@ -174,6 +174,7 @@ notebooks = [ "docs/tutorials/simulate-kicked-ising-tem.ipynb", "docs/tutorials/dc-hex-ising.ipynb", "docs/tutorials/ghz-spacetime-codes.ipynb", + "docs/tutorials/automated-spacetime-pauli-checks.ipynb", "docs/tutorials/sml-classification.ipynb", "docs/tutorials/qedma-2d-ising-with-qesem.ipynb", "docs/tutorials/readout-error-mitigation-sampler.ipynb", From 242167d0c1f2554e238e4c18f9e6d6b5b0cb0299 Mon Sep 17 00:00:00 2001 From: Henry Zou Date: Mon, 29 Jun 2026 20:14:06 -0400 Subject: [PATCH 02/14] Replace spacetime-codes tutorial with the qiskit-paulice workflow Rework the "Low-overhead error detection with spacetime codes" tutorial (docs/tutorials/ghz-spacetime-codes.ipynb) in place rather than adding a separate notebook. The tutorial now uses the qiskit-paulice package to automatically find and insert spacetime Pauli checks into Clifford circuits and boosts sampled-distribution fidelity by postselecting on check syndromes, replacing the previous hand-built-check / GHZ-preparation content. The title and URL (/docs/tutorials/ghz-spacetime-codes) are unchanged. - docs/tutorials/ghz-spacetime-codes.ipynb: new qiskit-paulice content, conforming to the tutorial template (Learning outcomes, Prerequisites, Background, Requirements, Setup, small-scale simulator example with Steps 1-4, large-scale hardware example, Next steps, References) - docs/tutorials/automated-spacetime-pauli-checks.ipynb: removed (the short-lived separate notebook); its extracted images moved to the ghz-spacetime-codes image folder - public/docs/images/tutorials/error-detection: removed (old GHZ outputs) - docs/tutorials/_toc.json, docs/tutorials/index.mdx: drop the extra entry; mark the refreshed tutorial isNew - qiskit_bot.yaml: add @henryzou50 as an owner of ghz-spacetime-codes - scripts/config/notebook-testing.toml: drop the extra exclusion entry The committed outputs are from a run on ibm_marrakesh, where the hardcoded ibm_boston layout maps onto bad edges and produces an unrealistic noise model (~19% gate noise). This breaks the demonstration: postselection does not improve fidelity and the postselection rate collapses to under 1%. Rerun on ibm_boston (or another Heron r3 backend) to restore a sane noise model and regenerate the outputs, then reconcile the prose (currently "Heron r3 QPU ibm_boston") and the usage estimate with the backend actually used. --- docs/tutorials/_toc.json | 6 +- .../automated-spacetime-pauli-checks.ipynb | 1113 --------- docs/tutorials/ghz-spacetime-codes.ipynb | 2226 ++++++----------- docs/tutorials/index.mdx | 2 - .../hardware-retrieve-1.avif | Bin 8560 -> 0 bytes .../extracted-outputs/step2-checks-1.avif | Bin 30037 -> 0 bytes .../extracted-outputs/step4-fidelity-1.avif | Bin 7034 -> 0 bytes .../extracted-outputs/step4-fidelity-2.avif | Bin 7786 -> 0 bytes .../extracted-outputs/step4-gamma-0.avif | Bin 8720 -> 0 bytes .../extracted-outputs/step4-psr-0.avif | Bin 6545 -> 0 bytes .../extracted-outputs/04bb6ea7-0.avif | Bin 1946 -> 0 bytes .../extracted-outputs/0c4b279b-0.avif | Bin 207210 -> 0 bytes .../extracted-outputs/246a42b8-0.avif | Bin 1656 -> 0 bytes .../extracted-outputs/4741910b-0.avif | Bin 2209 -> 0 bytes .../extracted-outputs/51a8f3d9-0.avif | Bin 3539 -> 0 bytes .../extracted-outputs/52add690-0.avif | Bin 3950 -> 0 bytes .../extracted-outputs/78c8c2b6-0.avif | Bin 61103 -> 0 bytes .../extracted-outputs/b905e875-1.avif | Bin 2146 -> 0 bytes .../extracted-outputs/d0faf114-0.avif | Bin 27667 -> 0 bytes .../extracted-outputs/d7902975-0.avif | Bin 13681 -> 0 bytes .../extracted-outputs/e1c1e16d-0.avif | Bin 1248 -> 0 bytes .../extracted-outputs/f8384a7d-0.avif | Bin 1656 -> 0 bytes .../hardware-retrieve-1.avif | Bin 0 -> 8965 bytes .../extracted-outputs/step1-code-0.avif | Bin .../extracted-outputs/step2-checks-1.avif | Bin 0 -> 29206 bytes .../extracted-outputs/step2-layout-1.avif | Bin .../extracted-outputs/step2-transpile-0.avif | Bin .../extracted-outputs/step4-fidelity-1.avif | Bin 0 -> 8181 bytes .../extracted-outputs/step4-fidelity-2.avif | Bin 0 -> 7997 bytes .../extracted-outputs/step4-gamma-0.avif | Bin 0 -> 8842 bytes .../extracted-outputs/step4-psr-0.avif | Bin 0 -> 6472 bytes qiskit_bot.yaml | 1 - 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"url": "/docs/tutorials/ghz-spacetime-codes" - }, - { - "title": "Automated spacetime Pauli checks with qiskit-paulice", - "url": "/docs/tutorials/automated-spacetime-pauli-checks", + "url": "/docs/tutorials/ghz-spacetime-codes", "isNew": true, "isNewDate": "2026-06-29" } diff --git a/docs/tutorials/automated-spacetime-pauli-checks.ipynb b/docs/tutorials/automated-spacetime-pauli-checks.ipynb deleted file mode 100644 index 5e67d537a19e..000000000000 --- a/docs/tutorials/automated-spacetime-pauli-checks.ipynb +++ /dev/null @@ -1,1113 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "id": "frontmatter", - "metadata": {}, - "source": [ - "---\n", - "title: Automated spacetime Pauli checks with qiskit-paulice\n", - "description: Automatically insert spacetime Pauli checks with qiskit-paulice and boost sampled Clifford-circuit fidelity by postselecting on check syndromes.\n", - "---\n", - "\n", - "# Automated spacetime Pauli checks with qiskit-paulice\n", - "\n", - "*Usage estimate: 1 minute on a Heron r3 processor (NOTE: This is an estimate only. Your runtime may vary.)*" - ] - }, - { - "cell_type": "markdown", - "id": "learning-outcomes", - "metadata": {}, - "source": [ - "## Learning outcomes\n", - "\n", - "After completing this tutorial, you can expect to understand the following information:\n", - "\n", - "- How spacetime Pauli checks detect logical errors in Clifford circuits, and how postselecting on their syndromes boosts the fidelity of a sampled distribution.\n", - "- How to use the `qiskit-paulice` package to find and insert hardware-efficient checks automatically with `get_check_qubits`, `NoiseModel`, and `add_pauli_checks`.\n", - "- How to estimate the fidelity of a stabilizer state by sampling its stabilizers and postselecting on check syndromes.\n", - "- How to run the full error-detection workflow on IBM Quantum® hardware and compare noisy and postselected fidelities." - ] - }, - { - "cell_type": "markdown", - "id": "prerequisites", - "metadata": {}, - "source": [ - "## Prerequisites\n", - "\n", - "It is recommended that you familiarize yourself with these topics:\n", - "\n", - "- The companion tutorial [Low-overhead error detection with spacetime codes](/docs/tutorials/ghz-spacetime-codes), which introduces coherent Pauli checks and builds a single check by hand. This tutorial picks up where that one leaves off and automates the same idea with `qiskit-paulice`.\n", - "- [Hardware fundamentals](/learning/courses/utility-scale-quantum-computing/hardware) for utility-scale quantum computing.\n", - "- The Clifford and stabilizer formalism, including how a stabilizer group describes a pure stabilizer state." - ] - }, - { - "cell_type": "markdown", - "id": "background", - "metadata": {}, - "source": [ - "## Background\n", - "\n", - "[Low-overhead error detection with spacetime codes](https://arxiv.org/abs/2504.15725) [[1]](#references) by Simon Martiel and Ali Javadi-Abhari introduces a method for detecting logical errors in Clifford-dominated circuits that sits between full error correction and lighter-weight error mitigation. The idea builds on coherent Pauli checks (CPC) from [Single-shot error mitigation by coherent Pauli checks](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033193) [[2]](#references) by van den Berg and others. In both approaches, a Clifford \"payload\" circuit is entangled with ancilla qubits to check certain invariants. Measuring the ancillas produces a syndrome that reports whether an error was detected during execution. Keeping only the samples with no detected error improves the fidelity of the sampled distribution, at the cost of a reduced postselection rate.\n", - "\n", - "The key difference between coherent Pauli checks and spacetime checks is the operators they measure. Coherent Pauli checks measure time-localized, high-weight operators. On qubit topologies with limited connectivity, such as heavy hex, those checks need many SWAP gates and often make the circuit too deep to run in practice. Implementing the checks as spacetime codes instead distributes each check across the payload circuit in space and time. This yields a hardware-efficient encoding that stays effective at detecting logical errors while keeping the qubit and depth overhead low.\n", - "\n", - "### What the qiskit-paulice package does\n", - "\n", - "The `qiskit-paulice` package automates the construction of these checks so that you do not have to build them by hand. Its primary role is to find and insert valid spacetime Pauli checks at the locations in a circuit that maximize error detection while minimizing qubit overhead. A check is *valid* when its operators leave the logical action of the payload circuit unchanged, *low-weight* when it uses few entangling gates, and *effective* when it detects a large fraction of the error relative to the noise the check itself introduces. The package scores candidate checks against a noise model and commits the best ones to the circuit. This tutorial uses three pieces of the API:\n", - "\n", - "- `get_check_qubits` inspects a backend coupling map and returns target and ancilla qubit pairs. A check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", - "- `NoiseModel.from_backend` builds a rough noise model from backend benchmark data. The model scores candidate checks, so an exact, learned noise model is not required. For a learned Pauli-Lindblad model, see `NoiseModel.from_pauli_lindblad_maps`.\n", - "- `add_pauli_checks` finds and inserts checks into a circuit. It returns a sequence of `CheckedCircuit` objects with an increasing number of checks, and each object provides a `get_postselection_method` that maps a measured bitstring to a syndrome vector. The `cost` argument selects the function that scores a check (`gamma`, the sampling overhead of the postselected inverse noise channel, or `LER`, the logical error rate). The `method` argument selects the search strategy (`windowed`, `genetic`, or `windowed_genetic`). This tutorial uses `cost=\"gamma\"` and `method=\"windowed\"`, which together give deterministic, reproducible check selection.\n", - "\n", - "### Estimating fidelity from stabilizer sampling\n", - "\n", - "To measure how well the error detection works, you can estimate the fidelity of the stabilizer state $|\\psi\\rangle = U|0\\rangle^{\\otimes n}$ that the circuit ideally prepares against the noisy state $\\rho$ that the hardware actually outputs. The projector onto a pure stabilizer state $|\\psi\\rangle$ equals the uniform average over the $2^n$ elements of its stabilizer group $\\mathcal{S}$:\n", - "\n", - "$$|\\psi\\rangle\\langle\\psi| = \\frac{1}{2^n}\\sum_{G \\in \\mathcal{S}} G.$$\n", - "\n", - "Substituting this into the fidelity gives the fidelity of $\\rho$ as the average expectation value of every stabilizer $G \\in \\mathcal{S}$ with respect to $\\rho$:\n", - "\n", - "$$F = \\mathrm{Tr}(\\rho|\\psi\\rangle\\langle\\psi|) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\mathrm{Tr}(\\rho G) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\langle G \\rangle_\\rho.$$\n", - "\n", - "For larger problems, enumerating all $2^n$ stabilizers is infeasible, so you can estimate the fidelity from a random sample. Drawing $M$ stabilizers $G_1, \\ldots, G_M$ uniformly at random from $\\mathcal{S}$ gives an unbiased estimator:\n", - "\n", - "$$\\hat F_M = \\frac{1}{M} \\sum_{i=1}^M \\langle G_i \\rangle_\\rho.$$\n", - "\n", - "Because a Clifford circuit prepares a stabilizer state, you can estimate its fidelity directly from sampled expectation values of its stabilizers. This tutorial first walks through the workflow on a simulator with a small circuit, then runs the same workflow on hardware with a deeper circuit. As circuits include more non-Clifford operations the number of valid checks shrinks quickly, so the method works best for Clifford-dominated circuits." - ] - }, - { - "cell_type": "markdown", - "id": "requirements", - "metadata": {}, - "source": [ - "## Requirements\n", - "\n", - "Before starting this tutorial, be sure you have the following installed:\n", - "\n", - "- Qiskit SDK v2.0 or later, with [visualization](/docs/api/qiskit/visualization) support\n", - "- Qiskit Runtime v0.40 or later (`pip install qiskit-ibm-runtime`)\n", - "- Qiskit Aer v0.17 or later (`pip install qiskit-aer`)\n", - "- Qiskit Paulice (`pip install qiskit-paulice`)\n", - "- tqdm (`pip install tqdm`)" - ] - }, - { - "cell_type": "markdown", - "id": "setup", - "metadata": {}, - "source": [ - "## Setup\n", - "\n", - "Import the required libraries and define the helper functions that are not available as imports. The `random_clifford_circuit` function builds a brickwork random Clifford payload, `append_basis_rotation` rotates a circuit so that a stabilizer is measured in the computational basis, `expectation` computes a stabilizer expectation value from sampled counts, and `cum_mean_sem` tracks the running fidelity estimate." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "id": "setup-imports", - "metadata": {}, - "outputs": [], - "source": [ - "# Standard library imports\n", - "import random\n", - "import time\n", - "\n", - "# External libraries\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from tqdm import tqdm\n", - "\n", - "# Qiskit\n", - "from qiskit import QuantumCircuit\n", - "from qiskit.quantum_info import Clifford, Pauli, PauliList\n", - "from qiskit.result import sampled_expectation_value\n", - "from qiskit.transpiler import generate_preset_pass_manager\n", - "from qiskit.visualization import plot_coupling_map\n", - "\n", - "# Qiskit Aer\n", - "from qiskit_aer import AerSimulator\n", - "from qiskit_aer.noise import NoiseModel as AerNoiseModel\n", - "from qiskit_aer.noise import ReadoutError, depolarizing_error\n", - "\n", - "# Qiskit IBM Runtime\n", - "from qiskit_ibm_runtime import QiskitRuntimeService\n", - "from qiskit_ibm_runtime import SamplerV2 as Sampler\n", - "\n", - "# Qiskit Paulice\n", - "from qiskit_paulice import add_pauli_checks\n", - "from qiskit_paulice.layout import get_check_qubits\n", - "from qiskit_paulice.noise_models import NoiseModel" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "id": "setup-helpers", - "metadata": {}, - "outputs": [], - "source": [ - "def random_clifford_circuit(\n", - " num_qubits: int, depth: int, rng: np.random.Generator\n", - ") -> QuantumCircuit:\n", - " \"\"\"Brickwork random Clifford on `num_qubits`, with `depth` CZ layers.\"\"\"\n", - " qc = QuantumCircuit(num_qubits)\n", - " qc.h(range(num_qubits))\n", - " for d in range(depth):\n", - " for i in range(d % 2, num_qubits - 1, 2):\n", - " qc.cz(i, i + 1)\n", - " for q in range(num_qubits):\n", - " if rng.integers(0, 2):\n", - " qc.sx(q)\n", - " if rng.integers(0, 2):\n", - " qc.s(q)\n", - " if rng.integers(0, 2):\n", - " qc.sx(q)\n", - " return qc\n", - "\n", - "\n", - "def append_basis_rotation(\n", - " circuit: QuantumCircuit, pauli: Pauli\n", - ") -> QuantumCircuit:\n", - " \"\"\"Strip measurements, append basis rotations for `pauli`, and re-measure.\"\"\"\n", - " out = circuit.remove_final_measurements(inplace=False)\n", - " for q in range(pauli.num_qubits):\n", - " if pauli.x[q]:\n", - " if pauli.z[q]:\n", - " out.sdg(q)\n", - " out.h(q)\n", - " out.measure_all()\n", - " return out\n", - "\n", - "\n", - "def expectation(counts: dict, pauli: Pauli) -> float:\n", - " \"\"\"Expectation value of `pauli` from counts measured in the Z basis.\n", - "\n", - " Pads with identity on any qubits beyond the support of `pauli`, such as the\n", - " check ancillas that appear in the postselected counts.\n", - " \"\"\"\n", - " if not counts:\n", - " return float(\"nan\")\n", - " n = pauli.num_qubits\n", - " sign = -1 if int(pauli.phase) % 4 == 2 else 1\n", - " total = len(next(iter(counts)))\n", - " label = \"\".join(\n", - " \"Z\" if q < n and (pauli.x[q] or pauli.z[q]) else \"I\"\n", - " for q in range(total - 1, -1, -1)\n", - " )\n", - " return sign * sampled_expectation_value(counts, label)\n", - "\n", - "\n", - "def cum_mean_sem(values: np.ndarray):\n", - " \"\"\"Cumulative mean and standard error of the mean, ignoring NaNs.\"\"\"\n", - " valid = ~np.isnan(values)\n", - " total = np.cumsum(np.where(valid, values, 0.0))\n", - " total_sq = np.cumsum(np.where(valid, values**2, 0.0))\n", - " count = np.maximum(np.cumsum(valid).astype(float), 1)\n", - " mean = total / count\n", - " sem = np.sqrt(np.maximum(total_sq / count - mean**2, 0) / count)\n", - " return np.where(np.cumsum(valid) > 0, mean, np.nan), sem" - ] - }, - { - "cell_type": "markdown", - "id": "small-scale-header", - "metadata": {}, - "source": [ - "## Small-scale simulator example\n", - "\n", - "This section walks through the full workflow on a noisy simulator. It uses backend benchmark data to choose a qubit layout and a noise model, finds checks automatically, and uses postselection on the sampled distribution to show the fidelity improvement." - ] - }, - { - "cell_type": "markdown", - "id": "step1-header", - "metadata": {}, - "source": [ - "### Step 1: Map classical inputs to a quantum problem\n", - "\n", - "The payload circuit is a shallow one-dimensional brickwork random Clifford circuit. Because the circuit is Clifford, it prepares a stabilizer state whose fidelity you can estimate directly from sampled stabilizer expectation values. Start with a shallow circuit so that the checks are easy to visualize in the next step." - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "id": "step1-code", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 3, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "num_qubits = 12\n", - "depth = 4\n", - "seed = 1764\n", - "rng = np.random.default_rng(seed)\n", - "np.random.seed(seed)\n", - "\n", - "circuit = random_clifford_circuit(num_qubits, depth, rng)\n", - "circuit.measure_all()\n", - "circuit.draw(\"mpl\", fold=-1, scale=0.6)" - ] - }, - { - "cell_type": "markdown", - "id": "step2-header", - "metadata": {}, - "source": [ - "### Step 2: Optimize problem for quantum hardware execution\n", - "\n", - "Mapping the circuit to hardware sets the physical qubit layout, the noise model that scores candidate checks, and the checks themselves.\n", - "\n", - "First, choose a one-dimensional qubit layout on the Heron r3 QPU `ibm_boston`. The layout zig-zags diagonally through the coupling graph to maximize the number of available ancilla qubits. The `get_check_qubits` function returns target and ancilla pairs, where a check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", - "\n", - "In the coupling graph that follows, the green qubits are payload qubits and the orange qubits are the ancillas that implement the checks. Qubits with an adjacent ancilla are used as target qubits for checks." - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "id": "step2-layout", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Target qubits: [68, 69, 89, 91, 111, 113, 133]\n", - "Ancilla qubits: [67, 70, 88, 92, 110, 114, 132]\n" - ] - }, - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 4, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "service = QiskitRuntimeService()\n", - "backend = service.backend(\"ibm_marrakesh\")\n", - "\n", - "# Choose a layout and pair each available target qubit with a neighboring ancilla\n", - "layout = [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133]\n", - "target_qubits, ancilla_qubits = get_check_qubits(backend, layout)\n", - "num_checks = len(target_qubits)\n", - "\n", - "print(f\"Target qubits: {target_qubits}\")\n", - "print(f\"Ancilla qubits: {ancilla_qubits}\")\n", - "plot_coupling_map(\n", - " num_qubits=backend.num_qubits,\n", - " qubit_coordinates=getattr(\n", - " backend.configuration(), \"qubit_coordinates\", None\n", - " ),\n", - " coupling_map=backend.configuration().coupling_map,\n", - " figsize=(12, 12),\n", - " qubit_color=[\n", - " \"#4CAF50\"\n", - " if i in set(layout)\n", - " else \"#FF9800\"\n", - " if i in set(ancilla_qubits)\n", - " else \"#DDDDDD\"\n", - " for i in backend.coupling_map.graph.node_indices()\n", - " ],\n", - " qubit_size=220,\n", - " line_width=2,\n", - " font_size=90,\n", - ")" - ] - }, - { - "cell_type": "markdown", - "id": "step2-transpile-md", - "metadata": {}, - "source": [ - "With the backend and layout chosen, transpile the payload into an instruction set architecture (ISA) circuit. Setting the layout and translating the gates into the native gate set of `ibm_boston` is enough here." - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "id": "step2-transpile", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 5, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "pm = generate_preset_pass_manager(\n", - " optimization_level=0, backend=backend, initial_layout=layout\n", - ")\n", - "circuit_isa = pm.run(circuit)\n", - "circuit_isa.draw(\"mpl\", fold=-1, scale=0.6)" - ] - }, - { - "cell_type": "markdown", - "id": "step2-noise-md", - "metadata": {}, - "source": [ - "Next, model how the gate and readout noise on the backend affects execution. The noise model determines where in the circuit a check captures the most error. A more accurate model improves the detection, but it is usually not necessary to learn one by sampling the QPU. The model that follows infers a uniform depolarizing channel for gate and readout noise from `qiskit-ibm-runtime` benchmark data." - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "id": "step2-noise", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "NoiseModel(gate_noise=0.19074977552838765, readout_noise=0.09235636393229167, idling_noise=None)\n" - ] - } - ], - "source": [ - "noise_model = NoiseModel.from_backend(\n", - " backend, layout, uniform_gate_noise=True\n", - ")\n", - "print(noise_model)" - ] - }, - { - "cell_type": "markdown", - "id": "step2-checks-md", - "metadata": {}, - "source": [ - "Now add checks to the circuit. The `add_pauli_checks` function takes the Clifford payload, the list of target qubits, and the noise model. The `ancilla_qubits` argument tells the function which physical ancilla to pair with each target. Checks are added in the order the target qubits appear, so the final layout of the checked circuit is `layout + ancilla_qubits`. To run an output circuit with fewer (`i`) checks, the final layout is `layout + ancilla_qubits[:i]`.\n", - "\n", - "The output of `add_pauli_checks` is a sequence of circuits with an increasing number of checks, from no checks up to one check on every target qubit. The visualization confirms that the checks use the specified target and ancilla pairs. For details on finding good checks, see Sections II to IV of the supplementary information in reference [[1]](#references)." - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "id": "step2-checks", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Physical layout of payload and ancillas: [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133, 67, 70, 88, 92, 110, 114, 132]\n", - "Checked circuit:\n" - ] - }, - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "checked = add_pauli_checks(\n", - " circuit_isa,\n", - " target_qubits,\n", - " noise_model,\n", - " ancilla_qubits=ancilla_qubits,\n", - " cost=\"gamma\",\n", - " method=\"windowed\",\n", - " seed=seed,\n", - ")\n", - "\n", - "print(f\"Physical layout of payload and ancillas: {layout + ancilla_qubits}\")\n", - "print(\"Checked circuit:\")\n", - "checked[-1].circuit.draw(\"mpl\", fold=-1, idle_wires=False)" - ] - }, - { - "cell_type": "markdown", - "id": "step3-header", - "metadata": {}, - "source": [ - "### Step 3: Execute using Qiskit primitives\n", - "\n", - "To make the effect of gate noise visible, increase the depth of the payload and sample a subset of its stabilizers. Each stabilizer is generally not qubit-wise commuting with the others, so a single set of checks is not valid for two different stabilizers. Rather than group stabilizers into commuting sets, find a good set of checks for each stabilizer independently. Sampling stabilizers uniformly at random gives an unbiased fidelity estimate.\n", - "\n", - "Build the deeper circuit and draw a random sample of its stabilizers." - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "id": "step3-stabilizers", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {-XYXZIZXYIIXZ, -XZYYIIXZXYYY, ...}\n" - ] - } - ], - "source": [ - "depth = 24\n", - "num_stabilizers = 20\n", - "num_shots = 1_000\n", - "\n", - "circuit = random_clifford_circuit(num_qubits, depth, rng)\n", - "\n", - "# Build the full stabilizer group, then sample from it uniformly at random\n", - "circ_no_meas = circuit.remove_final_measurements(inplace=False)\n", - "stabilizer_group = PauliList([Pauli(\"I\" * num_qubits)])\n", - "for generator in (\n", - " Pauli(label) for label in Clifford(circ_no_meas).to_labels(mode=\"S\")\n", - "):\n", - " stabilizer_group = stabilizer_group + stabilizer_group.compose(generator)\n", - "\n", - "keep = np.where(\n", - " stabilizer_group.x.any(axis=1) | stabilizer_group.z.any(axis=1)\n", - ")[0]\n", - "chosen = np.random.default_rng(seed).choice(\n", - " keep, size=min(num_stabilizers, len(keep)), replace=False\n", - ")\n", - "stabilizers = [stabilizer_group[int(i)] for i in chosen]\n", - "\n", - "two_qubit_depth = circuit.depth(lambda x: x.operation.num_qubits == 2)\n", - "print(\n", - " f\"Sampled {len(stabilizers)} stabilizers of a {circuit.num_qubits}-qubit \"\n", - " f\"circuit with two-qubit depth {two_qubit_depth}: \"\n", - " f\"{{{stabilizers[0]}, {stabilizers[1]}, ...}}\"\n", - ")" - ] - }, - { - "cell_type": "markdown", - "id": "step3-findchecks-md", - "metadata": {}, - "source": [ - "For each sampled stabilizer, rotate the circuit so that the stabilizer is measured in the computational basis, transpile it onto the backend, and find a good set of checks. The target and ancilla pairs are shuffled together for each stabilizer so that each target keeps its ancilla. Remember that checks are committed sequentially in the order the target qubits are given, and a committed check is not changed as more checks are added." - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "id": "step3-findchecks", - "metadata": {}, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "100%|██████████| 20/20 [00:15<00:00, 1.32it/s]" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Added 7 checks to 20 circuits in 15s.\n", - "On average, two-qubit depth increased from 24 to 32 when adding 7 checks.\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "\n" - ] - } - ], - "source": [ - "noisy_circuits = []\n", - "checked_circuits = []\n", - "depths_2q = []\n", - "t0 = time.time()\n", - "for i, pauli in enumerate(tqdm(stabilizers)):\n", - " noisy_circuits.append(pm.run(append_basis_rotation(circuit, pauli)))\n", - " # Shuffle target and ancilla pairs together so each target keeps its ancilla\n", - " targets, ancillas = zip(\n", - " *random.sample(\n", - " list(zip(target_qubits, ancilla_qubits, strict=True)),\n", - " k=len(target_qubits),\n", - " ),\n", - " strict=True,\n", - " )\n", - " checked_circuits.append(\n", - " add_pauli_checks(\n", - " noisy_circuits[-1],\n", - " list(targets),\n", - " noise_model,\n", - " ancilla_qubits=list(ancillas),\n", - " cost=\"gamma\",\n", - " method=\"windowed\",\n", - " seed=seed + 1 + i,\n", - " )\n", - " )\n", - " depths_2q.append(\n", - " checked_circuits[-1][-1].circuit.depth(lambda x: len(x.qubits) == 2)\n", - " )\n", - "\n", - "print(\n", - " f\"Added {num_checks} checks to {len(stabilizers)} circuits \"\n", - " f\"in {(time.time() - t0):.0f}s.\"\n", - ")\n", - "print(\n", - " f\"On average, two-qubit depth increased from \"\n", - " f\"{circuit.depth(lambda x: len(x.qubits) == 2)} to {int(np.mean(depths_2q))} \"\n", - " f\"when adding {num_checks} checks.\"\n", - ")" - ] - }, - { - "cell_type": "markdown", - "id": "step3-sample-md", - "metadata": {}, - "source": [ - "Sample the bare payload and the checked circuits with Qiskit Aer. The simulator uses the same depolarizing model that scored the checks, so the noise that the checks target is the noise that the simulator applies." - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "id": "step3-sample", - "metadata": {}, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "100%|██████████| 20/20 [00:18<00:00, 1.11it/s]\n" - ] - } - ], - "source": [ - "aer_nm = AerNoiseModel()\n", - "aer_nm.add_all_qubit_quantum_error(\n", - " depolarizing_error(noise_model.gate_noise, 2), [\"cz\"]\n", - ")\n", - "p = noise_model.readout_noise\n", - "aer_nm.add_all_qubit_readout_error(ReadoutError([[1 - p, p], [p, 1 - p]]))\n", - "noisy_sim = AerSimulator(method=\"stabilizer\", noise_model=aer_nm)\n", - "\n", - "counts = []\n", - "for i, checked_circ_result in enumerate(tqdm(checked_circuits)):\n", - " noisy_counts = (\n", - " noisy_sim.run(\n", - " noisy_circuits[i], shots=num_shots, seed_simulator=seed * i + 1\n", - " )\n", - " .result()\n", - " .get_counts()\n", - " )\n", - " checked_counts_per_variant = []\n", - " for k, ck in enumerate(checked_circ_result):\n", - " variant_counts = (\n", - " noisy_sim.run(\n", - " ck.circuit, shots=num_shots, seed_simulator=seed * i + 2 + k\n", - " )\n", - " .result()\n", - " .get_counts()\n", - " )\n", - " checked_counts_per_variant.append(variant_counts)\n", - " counts.append((noisy_counts, checked_counts_per_variant))" - ] - }, - { - "cell_type": "markdown", - "id": "step4-header", - "metadata": {}, - "source": [ - "### Step 4: Post-process and return result in desired classical format\n", - "\n", - "Each check uses entangling gates between one ancilla and one target. The ancilla starts in $|0\\rangle$, so $Z_\\text{anc}$ stabilizes its input. Propagating $Z_\\text{anc}$ forward through the checked circuit yields a Pauli operator on the output whose non-identity terms define the check's support. A check passes when the bits in its support have even parity. A sample is kept only when every check passes.\n", - "\n", - "The `get_postselection_method` of each `CheckedCircuit` returns a function that maps a measured bitstring to a syndrome vector. Keep the samples whose syndrome is zero for every check, and discard the rest. The chart that follows shows that adding more checks lowers the postselection rate. A lower postselection rate needs more shots to reach a target accuracy, so there is a tradeoff between detection capability and sampling cost. The rate appears to converge, which indicates that additional checks contribute less detection capability." - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "id": "step4-psr", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "rate_per_variant = []\n", - "kept_per_stab = []\n", - "for i, (_, checked_counts_per_variant) in enumerate(counts):\n", - " rates = []\n", - " kept_at_num_checks = None\n", - " for k, variant_counts in enumerate(checked_counts_per_variant):\n", - " ps_fn = checked_circuits[i][k].get_postselection_method()\n", - " kept = {\n", - " bs: n for bs, n in variant_counts.items() if not ps_fn(bs).any()\n", - " }\n", - " rates.append(sum(kept.values()) / num_shots)\n", - " if k == num_checks:\n", - " kept_at_num_checks = kept\n", - " rate_per_variant.append(rates)\n", - " kept_per_stab.append(kept_at_num_checks)\n", - "\n", - "max_len = max(len(s) for s in rate_per_variant)\n", - "rates_arr = np.full((len(rate_per_variant), max_len), np.nan)\n", - "for i, s in enumerate(rate_per_variant):\n", - " rates_arr[i, : len(s)] = s\n", - "ks = np.arange(max_len)\n", - "\n", - "fig, ax = plt.subplots(figsize=(8, 4))\n", - "ax.plot(ks, rates_arr.T, color=\"#ff8c00\", alpha=0.15, linewidth=1)\n", - "ax.plot(\n", - " ks,\n", - " np.nanmedian(rates_arr, axis=0),\n", - " color=\"black\",\n", - " linewidth=1,\n", - " linestyle=\"--\",\n", - " label=\"median\",\n", - ")\n", - "ax.set_xlabel(\"Checks committed\")\n", - "ax.set_ylabel(\"Postselection rate\")\n", - "ax.set_ylim((0, 1.05))\n", - "ax.set_title(\n", - " f\"Per-stabilizer postselection rate ({len(rates_arr)} stabilizers)\"\n", - ")\n", - "ax.legend()\n", - "ax.grid(True, alpha=0.3)\n", - "plt.tight_layout()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "id": "step4-fidelity-md", - "metadata": {}, - "source": [ - "Now compare the fidelity of the bare noisy state with the postselected state. Postselecting only the samples with no detected error raises the expectation value of every stabilizer, and therefore the estimated fidelity. The postselected values use fewer samples than the raw values, yet the expectation values are more accurate and the sampled variance is lower." - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "id": "step4-fidelity", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "ideal fidelity: 1.0\n", - "noisy fidelity: -0.0058\n", - "postselected fidelity: -0.1257\n", - "mean postselection rate: 0.007\n" - ] - }, - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "results = []\n", - "for i, ((noisy_counts, _), kept) in enumerate(\n", - " zip(counts, kept_per_stab, strict=True)\n", - "):\n", - " results.append(\n", - " (\n", - " expectation(noisy_counts, stabilizers[i]),\n", - " expectation(kept, stabilizers[i]),\n", - " sum(kept.values()) / num_shots,\n", - " )\n", - " )\n", - "\n", - "fidelity_noisy = float(np.nanmean([r[0] for r in results]))\n", - "fidelity_postsel = float(np.nanmean([r[1] for r in results]))\n", - "psr = float(np.mean([r[2] for r in results]))\n", - "print(\n", - " f\"ideal fidelity: 1.0\\n\"\n", - " f\"noisy fidelity: {fidelity_noisy:.4f}\\n\"\n", - " f\"postselected fidelity: {fidelity_postsel:.4f}\\n\"\n", - " f\"mean postselection rate: {psr:.3f}\"\n", - ")\n", - "\n", - "evs_ideal = np.ones(len(results))\n", - "evs_noisy = np.array([r[0] for r in results])\n", - "evs_post = np.array([r[1] for r in results])\n", - "idx = np.arange(len(results))\n", - "\n", - "\n", - "def strip(ax, ys, color, label):\n", - " m, s = np.nanmean(ys), np.nanstd(ys)\n", - " ax.axhspan(\n", - " m - s, m + s, color=color, alpha=0.15, label=f\"{label} mean and std\"\n", - " )\n", - " ax.axhline(m, color=color, linewidth=1, linestyle=\"--\")\n", - "\n", - "\n", - "fig, ax = plt.subplots(figsize=(8, 4))\n", - "ax.axhline(np.nanmean(evs_ideal), color=\"black\", linewidth=1.5, label=\"ideal\")\n", - "strip(ax, evs_noisy, \"red\", \"noisy\")\n", - "strip(ax, evs_post, \"green\", \"postselected\")\n", - "ax.scatter(idx, evs_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", - "ax.scatter(\n", - " idx,\n", - " evs_post,\n", - " color=\"green\",\n", - " s=22,\n", - " alpha=0.7,\n", - " label=\"postselected\",\n", - ")\n", - "ax.set_xlabel(\"stabilizer index\")\n", - "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", - "ax.set_ylim((-0.1, 1.1))\n", - "ax.set_title(\"Per-stabilizer expectation values\")\n", - "ax.legend(loc=\"lower left\")\n", - "ax.grid(True, alpha=0.3)\n", - "plt.tight_layout()\n", - "plt.show()\n", - "\n", - "M = np.arange(1, len(results) + 1)\n", - "fig, ax = plt.subplots(figsize=(8, 4))\n", - "for ys, color, label in [\n", - " (evs_ideal, \"black\", \"ideal\"),\n", - " (evs_noisy, \"red\", \"noisy\"),\n", - " (evs_post, \"green\", \"postselected\"),\n", - "]:\n", - " cm, sem = cum_mean_sem(ys)\n", - " ax.plot(M, cm, color=color, linewidth=1.5, label=label)\n", - " ax.fill_between(M, cm - sem, cm + sem, color=color, alpha=0.15)\n", - "ax.set_xlabel(\"number of stabilizers averaged\")\n", - "ax.set_ylabel(\"running fidelity estimate\")\n", - "ax.set_title(\"Fidelity convergence versus number of stabilizers\")\n", - "ax.legend()\n", - "ax.grid(True, alpha=0.3)\n", - "plt.tight_layout()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "id": "step4-gamma-md", - "metadata": {}, - "source": [ - "The gamma score reports how much of the modeled noise channel remains undetected by the checks. Plotting the gamma score against the number of committed checks shows how the detection capability improves as each check is added. A gamma curve that converges toward `1.0` indicates that additional checks detect little extra error, which matches the convergence seen in the postselection rate." - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "id": "step4-gamma", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "stab_scores = [\n", - " [variant.cost for variant in checked_circ_result]\n", - " for checked_circ_result in checked_circuits\n", - "]\n", - "max_len = max(len(s) for s in stab_scores)\n", - "scores = np.full((len(stab_scores), max_len), np.nan)\n", - "for i, s in enumerate(stab_scores):\n", - " scores[i, : len(s)] = s\n", - "ks = np.arange(max_len)\n", - "\n", - "fig, ax = plt.subplots(figsize=(8, 4))\n", - "ax.plot(ks, scores.T, color=\"#4682b4\", alpha=0.15, linewidth=1)\n", - "ax.plot(\n", - " ks,\n", - " np.nanmedian(scores, axis=0),\n", - " color=\"black\",\n", - " linewidth=1,\n", - " linestyle=\"--\",\n", - " label=\"median\",\n", - ")\n", - "ax.set_xlabel(\"Checks committed\")\n", - "ax.set_ylabel(\"Gamma\")\n", - "ax.set_yscale(\"log\")\n", - "ax.set_title(f\"Per-stabilizer gamma curves ({len(scores)} stabilizers)\")\n", - "ax.legend()\n", - "ax.grid(True, alpha=0.3, which=\"both\")\n", - "plt.tight_layout()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "id": "hardware-header", - "metadata": {}, - "source": [ - "## Large-scale hardware example\n", - "\n", - "The same workflow runs on hardware with a deeper payload and more stabilizers. This section reuses the backend, layout, target and ancilla pairs, noise model, and pass manager from the simulator example, then submits the sampled circuits to the QPU in a single job. Because the circuit is deeper, gate noise has a larger effect, and the fidelity gain from postselection is more pronounced.\n", - "\n", - "The parameters that follow set the depth, the number of stabilizers, and the number of shots. Increase `hw_num_stabilizers` for a more precise fidelity estimate at the cost of more circuits per job, and adjust the layout to use more qubits if you want a larger payload." - ] - }, - { - "cell_type": "markdown", - "id": "hardware-steps-header", - "metadata": {}, - "source": [ - "### Steps 1-4 (compressed into a single code block)\n", - "\n", - "The first cell builds the deeper payload, samples its stabilizers, finds the fully checked circuit for each one, and submits one Sampler job that contains both the bare and the checked circuits. Each job carries the tag `TUT_ASPC` so that you can find it later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "id": "hardware-submit", - "metadata": {}, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "100%|██████████| 50/50 [03:33<00:00, 4.27s/it]\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Submitted job d919t1fqq29s738n85cg with 100 circuits\n" - ] - } - ], - "source": [ - "# -------------------------Step 1: build a deeper payload and sample its stabilizers-------------------------\n", - "hw_depth = 72\n", - "hw_num_stabilizers = 50\n", - "hw_num_shots = 2_000\n", - "\n", - "hw_circuit = random_clifford_circuit(num_qubits, hw_depth, rng)\n", - "hw_no_meas = hw_circuit.remove_final_measurements(inplace=False)\n", - "hw_group = PauliList([Pauli(\"I\" * num_qubits)])\n", - "for generator in (\n", - " Pauli(label) for label in Clifford(hw_no_meas).to_labels(mode=\"S\")\n", - "):\n", - " hw_group = hw_group + hw_group.compose(generator)\n", - "keep = np.where(hw_group.x.any(axis=1) | hw_group.z.any(axis=1))[0]\n", - "chosen = np.random.default_rng(seed).choice(\n", - " keep, size=min(hw_num_stabilizers, len(keep)), replace=False\n", - ")\n", - "hw_stabilizers = [hw_group[int(i)] for i in chosen]\n", - "\n", - "# -------------------------Step 2: transpile and add the fully checked circuit per stabilizer-------------------------\n", - "hw_noisy_circuits = []\n", - "hw_checked_circuits = []\n", - "for i, pauli in enumerate(tqdm(hw_stabilizers)):\n", - " bare = pm.run(append_basis_rotation(hw_circuit, pauli))\n", - " hw_noisy_circuits.append(bare)\n", - " variants = add_pauli_checks(\n", - " bare,\n", - " target_qubits,\n", - " noise_model,\n", - " ancilla_qubits=ancilla_qubits,\n", - " cost=\"gamma\",\n", - " method=\"windowed\",\n", - " seed=seed + 1 + i,\n", - " )\n", - " hw_checked_circuits.append(variants[-1]) # keep the fully checked circuit\n", - "\n", - "# -------------------------Step 3: submit one Sampler job with the bare and checked circuits-------------------------\n", - "sampler = Sampler(mode=backend)\n", - "sampler.options.default_shots = hw_num_shots\n", - "sampler.options.environment.job_tags = [\"TUT_ASPC\"]\n", - "\n", - "pubs = hw_noisy_circuits + [cc.circuit for cc in hw_checked_circuits]\n", - "job = sampler.run(pubs)\n", - "print(f\"Submitted job {job.job_id()} with {len(pubs)} circuits\")" - ] - }, - { - "cell_type": "markdown", - "id": "hardware-retrieve-md", - "metadata": {}, - "source": [ - "After the job completes, retrieve the results and postselect the checked counts. Compute the noisy and postselected fidelities as the average stabilizer expectation value over the sampled stabilizers." - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "id": "hardware-retrieve", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "noisy fidelity: 0.0000\n", - "postselected fidelity: 0.0125\n", - "mean postselection rate: 0.008\n" - ] - }, - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "# -------------------------Step 4: postselect and compare fidelity-------------------------\n", - "result = job.result()\n", - "n_stab = len(hw_stabilizers)\n", - "\n", - "hw_results = []\n", - "for i in range(n_stab):\n", - " noisy_counts = result[i].join_data().get_counts()\n", - " checked_counts = result[n_stab + i].join_data().get_counts()\n", - " ps_fn = hw_checked_circuits[i].get_postselection_method()\n", - " kept = {bs: c for bs, c in checked_counts.items() if not ps_fn(bs).any()}\n", - " hw_results.append(\n", - " (\n", - " expectation(noisy_counts, hw_stabilizers[i]),\n", - " expectation(kept, hw_stabilizers[i]),\n", - " sum(kept.values()) / sum(checked_counts.values()),\n", - " )\n", - " )\n", - "\n", - "hw_fidelity_noisy = float(np.nanmean([r[0] for r in hw_results]))\n", - "hw_fidelity_postsel = float(np.nanmean([r[1] for r in hw_results]))\n", - "hw_psr = float(np.mean([r[2] for r in hw_results]))\n", - "print(\n", - " f\"noisy fidelity: {hw_fidelity_noisy:.4f}\\n\"\n", - " f\"postselected fidelity: {hw_fidelity_postsel:.4f}\\n\"\n", - " f\"mean postselection rate: {hw_psr:.3f}\"\n", - ")\n", - "\n", - "hw_noisy = np.array([r[0] for r in hw_results])\n", - "hw_post = np.array([r[1] for r in hw_results])\n", - "idx = np.arange(n_stab)\n", - "\n", - "fig, ax = plt.subplots(figsize=(8, 4))\n", - "ax.axhline(1.0, color=\"black\", linewidth=1.5, label=\"ideal\")\n", - "strip(ax, hw_noisy, \"red\", \"noisy\")\n", - "strip(ax, hw_post, \"green\", \"postselected\")\n", - "ax.scatter(idx, hw_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", - "ax.scatter(\n", - " idx,\n", - " hw_post,\n", - " color=\"green\",\n", - " s=22,\n", - " alpha=0.7,\n", - " label=\"postselected\",\n", - ")\n", - "ax.set_xlabel(\"stabilizer index\")\n", - "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", - "ax.set_ylim((-0.1, 1.1))\n", - "ax.set_title(\"Per-stabilizer expectation values on hardware\")\n", - "ax.legend(loc=\"lower left\")\n", - "ax.grid(True, alpha=0.3)\n", - "plt.tight_layout()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "id": "next-steps", - "metadata": {}, - "source": [ - "## Next steps\n", - "\n", - "\n", - "If you found this work interesting, you might be interested in the following material:\n", - "\n", - "- The conceptual companion tutorial [Low-overhead error detection with spacetime codes](/docs/tutorials/ghz-spacetime-codes), which builds a coherent Pauli check by hand and prepares a GHZ state on hardware.\n", - "- The tutorial on [repetition codes](/docs/tutorials/repetition-codes) for an introduction to quantum error correction.\n", - "- The [`qiskit-paulice` package](https://github.com/Qiskit/qiskit-paulice) for the full check-finding API.\n", - "- The paper [Low-overhead error detection with spacetime codes](https://arxiv.org/abs/2504.15725) for the theory behind the checks.\n", - "" - ] - }, - { - "cell_type": "markdown", - "id": "references", - "metadata": {}, - "source": [ - "## References\n", - "\n", - "- [1] Martiel, S., & Javadi-Abhari, A. (2025). Low-overhead error detection with spacetime codes. *arXiv preprint* [arXiv:2504.15725](https://arxiv.org/abs/2504.15725).\n", - "- [2] van den Berg, E., Bravyi, S., Gambetta, J. M., Jurcevic, P., Maslov, D., & Temme, K. (2023). Single-shot error mitigation by coherent Pauli checks. *Physical Review Research*, 5(3), 033193." - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index ff1e2e3fc626..fa4bfdb49d37 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -2,250 +2,354 @@ "cells": [ { "cell_type": "markdown", - "id": "59c663c5", + "id": "frontmatter", "metadata": {}, "source": [ "---\n", "title: Low-overhead error detection with spacetime codes\n", - "description: Experiment with error detection on a small-scale random Clifford circuit, then walk through the task of GHZ state preparation.\n", + "description: Automatically insert spacetime Pauli checks with qiskit-paulice and boost sampled Clifford-circuit fidelity by postselecting on check syndromes.\n", "---\n", "\n", - "\n", - "{/* cspell:ignore unflagged Tunables parallelizable uncomputation */}\n", - "\n", "# Low-overhead error detection with spacetime codes\n", "\n", - "*Usage estimate: 10 seconds on a Heron r3 processor (NOTE: This is an estimate only. Your runtime might vary.)*\n", - "\n", + "*Usage estimate: 1 minute on a Heron r3 processor (NOTE: This is an estimate only. Your runtime may vary.)*" + ] + }, + { + "cell_type": "markdown", + "id": "learning-outcomes", + "metadata": {}, + "source": [ "## Learning outcomes\n", "\n", - "After going through this tutorial, users should understand the following:\n", - "- The basics of setting up an error detection technique using spacetime codes\n", - "- How to set up a high-fidelity GHZ state on hardware using error detection\n", + "After completing this tutorial, you can expect to understand the following information:\n", "\n", + "- How spacetime Pauli checks detect logical errors in Clifford circuits, and how postselecting on their syndromes boosts the fidelity of a sampled distribution.\n", + "- How to use the `qiskit-paulice` package to find and insert hardware-efficient checks automatically with `get_check_qubits`, `NoiseModel`, and `add_pauli_checks`.\n", + "- How to estimate the fidelity of a stabilizer state by sampling its stabilizers and postselecting on check syndromes.\n", + "- How to run the full error-detection workflow on IBM Quantum® hardware and compare noisy and postselected fidelities." + ] + }, + { + "cell_type": "markdown", + "id": "prerequisites", + "metadata": {}, + "source": [ "## Prerequisites\n", "\n", - "It is recommended that you familiarize yourself with this topic:\n", - "- [Hardware fundamentals](/learning/courses/utility-scale-quantum-computing/hardware) for utility-scale quantum computing\n", + "It is recommended that you familiarize yourself with these topics:\n", "\n", + "- [Hardware fundamentals](/learning/courses/utility-scale-quantum-computing/hardware) for utility-scale quantum computing.\n", + "- The Clifford and stabilizer formalism, including how a stabilizer group describes a pure stabilizer state." + ] + }, + { + "cell_type": "markdown", + "id": "background", + "metadata": {}, + "source": [ "## Background\n", "\n", - "[Low-overhead error detection with spacetime codes](https://arxiv.org/abs/2504.15725) [[1]](#references) by Simon Martiel and Ali Javadi-Abhari proposes synthesizing low-weight, connectivity-aware spacetime checks for Clifford-dominated circuits, then postselecting on those checks to catch faults with far less overhead than full error correction and fewer shots than standard error mitigation.\n", + "[Low-overhead error detection with spacetime codes](https://arxiv.org/abs/2504.15725) [[1]](#references) by Simon Martiel and Ali Javadi-Abhari introduces a method for detecting logical errors in Clifford-dominated circuits that sits between full error correction and lighter-weight error mitigation. The idea builds on coherent Pauli checks (CPC) from [Single-shot error mitigation by coherent Pauli checks](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033193) [[2]](#references) by van den Berg and others. In both approaches, a Clifford \"payload\" circuit is entangled with ancilla qubits to check certain invariants. Measuring the ancillas produces a syndrome that reports whether an error was detected during execution. Keeping only the samples with no detected error improves the fidelity of the sampled distribution, at the cost of a reduced postselection rate.\n", + "\n", + "The key difference between coherent Pauli checks and spacetime checks is the operators they measure. Coherent Pauli checks measure time-localized, high-weight operators. On qubit topologies with limited connectivity, such as heavy hex, those checks need many SWAP gates and often make the circuit too deep to run in practice. Implementing the checks as spacetime codes instead distributes each check across the payload circuit in space and time. This yields a hardware-efficient encoding that stays effective at detecting logical errors while keeping the qubit and depth overhead low.\n", + "\n", + "### What the qiskit-paulice package does\n", + "\n", + "The `qiskit-paulice` package automates the construction of these checks so that you do not have to build them by hand. Its primary role is to find and insert valid spacetime Pauli checks at the locations in a circuit that maximize error detection while minimizing qubit overhead. A check is *valid* when its operators leave the logical action of the payload circuit unchanged, *low-weight* when it uses few entangling gates, and *effective* when it detects a large fraction of the error relative to the noise the check itself introduces. The package scores candidate checks against a noise model and commits the best ones to the circuit. This tutorial uses three pieces of the API:\n", "\n", - "This paper proposes a novel method for error detection in quantum circuits (specifically Clifford circuits) that strikes a balance between full error correction and lighter-weight mitigation techniques. The key idea is to use spacetime codes to generate \"checks\" across the circuit that are capable of catching errors, with significantly lower qubit and gate overhead than full fault-tolerant error correction. The authors design efficient algorithms to select checks that are low-weight (involving few qubits), compatible with the physical connectivity of the device, and cover large temporal and spatial regions of the circuit. They demonstrate the approach on circuits with up to 50 logical qubits and ~2450 CZ gates, achieving physical-to-logical fidelity gains of up to 236x. Also note that as circuits include more non-Clifford operations, the number of valid checks shrinks exponentially, indicating the method works best for Clifford-dominated circuits. Overall, in the near term, error detection via spacetime codes may offer a practical, lower-overhead route to improving reliability in quantum hardware.\n", + "- `get_check_qubits` inspects a backend coupling map and returns target and ancilla qubit pairs. A check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", + "- `NoiseModel.from_backend` builds a rough noise model from backend benchmark data. The model scores candidate checks, so an exact, learned noise model is not required. For a learned Pauli-Lindblad model, see `NoiseModel.from_pauli_lindblad_maps`.\n", + "- `add_pauli_checks` finds and inserts checks into a circuit. It returns a sequence of `CheckedCircuit` objects with an increasing number of checks, and each object provides a `get_postselection_method` that maps a measured bitstring to a syndrome vector. The `cost` argument selects the function that scores a check (`gamma`, the sampling overhead of the postselected inverse noise channel, or `LER`, the logical error rate). The `method` argument selects the search strategy (`windowed`, `genetic`, or `windowed_genetic`). This tutorial uses `cost=\"gamma\"` and `method=\"windowed\"`, which together give deterministic, reproducible check selection.\n", "\n", - "This error detection technique relies on the notion of coherent Pauli checks and is based on the work [Single-shot error mitigation by coherent Pauli checks](https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033193) [[2]](#references) by van den Berg et al.\n", + "### Estimating fidelity from stabilizer sampling\n", "\n", - "More recently, the paper [Big cats: entanglement in 120 qubits and beyond](https://arxiv.org/abs/2510.09520) [[3]](#references) by Javadi-Abhari et al., reports the creation of a 120-qubit Greenberger-Horne-Zeilinger (GHZ) state, the largest multipartite entangled state achieved to date on a superconducting-qubit platform. Using a hardware-aware compiler, low-overhead error detection, and a \"temporary uncomputation\" technique to reduce noise, the researchers achieved a fidelity of 0.56 ± 0.03 with about 28% post-selection efficiency. The work demonstrates genuine entanglement across all 120 qubits, validating multiple fidelity-certification methods, and marks a major benchmark for scalable quantum hardware.\n", + "To measure how well the error detection works, you can estimate the fidelity of the stabilizer state $|\\psi\\rangle = U|0\\rangle^{\\otimes n}$ that the circuit ideally prepares against the noisy state $\\rho$ that the hardware actually outputs. The projector onto a pure stabilizer state $|\\psi\\rangle$ equals the uniform average over the $2^n$ elements of its stabilizer group $\\mathcal{S}$:\n", "\n", - "This tutorial builds on these ideas, guiding you through implementing the error detection algorithm first on a small-scale random Clifford circuit and then through the task of GHZ state preparation, to help you experiment with error detection on your own quantum circuits." + "$$|\\psi\\rangle\\langle\\psi| = \\frac{1}{2^n}\\sum_{G \\in \\mathcal{S}} G.$$\n", + "\n", + "Substituting this into the fidelity gives the fidelity of $\\rho$ as the average expectation value of every stabilizer $G \\in \\mathcal{S}$ with respect to $\\rho$:\n", + "\n", + "$$F = \\mathrm{Tr}(\\rho|\\psi\\rangle\\langle\\psi|) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\mathrm{Tr}(\\rho G) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\langle G \\rangle_\\rho.$$\n", + "\n", + "For larger problems, enumerating all $2^n$ stabilizers is infeasible, so you can estimate the fidelity from a random sample. Drawing $M$ stabilizers $G_1, \\ldots, G_M$ uniformly at random from $\\mathcal{S}$ gives an unbiased estimator:\n", + "\n", + "$$\\hat F_M = \\frac{1}{M} \\sum_{i=1}^M \\langle G_i \\rangle_\\rho.$$\n", + "\n", + "Because a Clifford circuit prepares a stabilizer state, you can estimate its fidelity directly from sampled expectation values of its stabilizers. This tutorial first walks through the workflow on a simulator with a small circuit, then runs the same workflow on hardware with a deeper circuit. As circuits include more non-Clifford operations the number of valid checks shrinks quickly, so the method works best for Clifford-dominated circuits." ] }, { "cell_type": "markdown", - "id": "fce0de6c", + "id": "requirements", "metadata": {}, "source": [ "## Requirements\n", + "\n", "Before starting this tutorial, be sure you have the following installed:\n", "\n", "- Qiskit SDK v2.0 or later, with [visualization](/docs/api/qiskit/visualization) support\n", "- Qiskit Runtime v0.40 or later (`pip install qiskit-ibm-runtime`)\n", - "- Qiskit Aer v0.17.2 (`pip install qiskit-aer`)" + "- Qiskit Aer v0.17 or later (`pip install qiskit-aer`)\n", + "- Qiskit Paulice (`pip install qiskit-paulice`)\n", + "- tqdm (`pip install tqdm`)" ] }, { "cell_type": "markdown", - "id": "4286334d", + "id": "setup", "metadata": {}, "source": [ - "## Setup" + "## Setup\n", + "\n", + "Import the required libraries and define the helper functions that are not available as imports. The `random_clifford_circuit` function builds a brickwork random Clifford payload, `append_basis_rotation` rotates a circuit so that a stabilizer is measured in the computational basis, `expectation` computes a stabilizer expectation value from sampled counts, and `cum_mean_sem` tracks the running fidelity estimate." ] }, { "cell_type": "code", - "execution_count": null, - "id": "8fdec072", + "execution_count": 1, + "id": "setup-imports", "metadata": {}, "outputs": [], "source": [ "# Standard library imports\n", - "from collections import defaultdict, deque\n", - "from functools import partial\n", + "import random\n", + "import time\n", "\n", "# External libraries\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", + "from tqdm import tqdm\n", "\n", "# Qiskit\n", - "from qiskit import ClassicalRegister, QuantumCircuit\n", - "from qiskit.circuit import Delay\n", - "from qiskit.circuit.library import RZGate, XGate\n", - "from qiskit.converters import circuit_to_dag, dag_to_circuit\n", - "from qiskit.quantum_info import Pauli, random_clifford\n", - "from qiskit.transpiler import AnalysisPass, PassManager\n", - "from qiskit.transpiler.passes import (\n", - " ALAPScheduleAnalysis,\n", - " CollectAndCollapse,\n", - " PadDelay,\n", - " PadDynamicalDecoupling,\n", - " RemoveBarriers,\n", - ")\n", - "from qiskit.transpiler.passes.optimization.collect_and_collapse import (\n", - " collect_using_filter_function,\n", - " collapse_to_operation,\n", - ")\n", - "from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager\n", - "from qiskit.visualization import plot_gate_map, plot_histogram\n", + "from qiskit import QuantumCircuit\n", + "from qiskit.quantum_info import Clifford, Pauli, PauliList\n", + "from qiskit.result import sampled_expectation_value\n", + "from qiskit.transpiler import generate_preset_pass_manager\n", + "from qiskit.visualization import plot_coupling_map\n", "\n", "# Qiskit Aer\n", "from qiskit_aer import AerSimulator\n", - "from qiskit_aer.noise import NoiseModel, ReadoutError, depolarizing_error\n", + "from qiskit_aer.noise import NoiseModel as AerNoiseModel\n", + "from qiskit_aer.noise import ReadoutError, depolarizing_error\n", "\n", "# Qiskit IBM Runtime\n", "from qiskit_ibm_runtime import QiskitRuntimeService\n", - "from qiskit_ibm_runtime import SamplerV2 as Sampler" + "from qiskit_ibm_runtime import SamplerV2 as Sampler\n", + "\n", + "# Qiskit Paulice\n", + "from qiskit_paulice import add_pauli_checks\n", + "from qiskit_paulice.layout import get_check_qubits\n", + "from qiskit_paulice.noise_models import NoiseModel" ] }, { - "cell_type": "markdown", - "id": "3888d8aa", + "cell_type": "code", + "execution_count": 2, + "id": "setup-helpers", "metadata": {}, + "outputs": [], "source": [ - "## Small-scale simulator example" + "def random_clifford_circuit(\n", + " num_qubits: int, depth: int, rng: np.random.Generator\n", + ") -> QuantumCircuit:\n", + " \"\"\"Brickwork random Clifford on `num_qubits`, with `depth` CZ layers.\"\"\"\n", + " qc = QuantumCircuit(num_qubits)\n", + " qc.h(range(num_qubits))\n", + " for d in range(depth):\n", + " for i in range(d % 2, num_qubits - 1, 2):\n", + " qc.cz(i, i + 1)\n", + " for q in range(num_qubits):\n", + " if rng.integers(0, 2):\n", + " qc.sx(q)\n", + " if rng.integers(0, 2):\n", + " qc.s(q)\n", + " if rng.integers(0, 2):\n", + " qc.sx(q)\n", + " return qc\n", + "\n", + "\n", + "def append_basis_rotation(\n", + " circuit: QuantumCircuit, pauli: Pauli\n", + ") -> QuantumCircuit:\n", + " \"\"\"Strip measurements, append basis rotations for `pauli`, and re-measure.\"\"\"\n", + " out = circuit.remove_final_measurements(inplace=False)\n", + " for q in range(pauli.num_qubits):\n", + " if pauli.x[q]:\n", + " if pauli.z[q]:\n", + " out.sdg(q)\n", + " out.h(q)\n", + " out.measure_all()\n", + " return out\n", + "\n", + "\n", + "def expectation(counts: dict, pauli: Pauli) -> float:\n", + " \"\"\"Expectation value of `pauli` from counts measured in the Z basis.\n", + "\n", + " Pads with identity on any qubits beyond the support of `pauli`, such as the\n", + " check ancillas that appear in the postselected counts.\n", + " \"\"\"\n", + " if not counts:\n", + " return float(\"nan\")\n", + " n = pauli.num_qubits\n", + " sign = -1 if int(pauli.phase) % 4 == 2 else 1\n", + " total = len(next(iter(counts)))\n", + " label = \"\".join(\n", + " \"Z\" if q < n and (pauli.x[q] or pauli.z[q]) else \"I\"\n", + " for q in range(total - 1, -1, -1)\n", + " )\n", + " return sign * sampled_expectation_value(counts, label)\n", + "\n", + "\n", + "def cum_mean_sem(values: np.ndarray):\n", + " \"\"\"Cumulative mean and standard error of the mean, ignoring NaNs.\"\"\"\n", + " valid = ~np.isnan(values)\n", + " total = np.cumsum(np.where(valid, values, 0.0))\n", + " total_sq = np.cumsum(np.where(valid, values**2, 0.0))\n", + " count = np.maximum(np.cumsum(valid).astype(float), 1)\n", + " mean = total / count\n", + " sem = np.sqrt(np.maximum(total_sq / count - mean**2, 0) / count)\n", + " return np.where(np.cumsum(valid) > 0, mean, np.nan), sem" ] }, { "cell_type": "markdown", - "id": "31bd3de5", + "id": "small-scale-header", "metadata": {}, "source": [ - "### Step 1: Map classical inputs to a quantum problem" + "## Small-scale simulator example\n", + "\n", + "This section walks through the full workflow on a noisy simulator. It uses backend benchmark data to choose a qubit layout and a noise model, finds checks automatically, and uses postselection on the sampled distribution to show the fidelity improvement." ] }, { "cell_type": "markdown", - "id": "704eb75f", + "id": "step1-header", "metadata": {}, "source": [ - "To demonstrate this method, we start by constructing a simple Clifford circuit. Our goal is to be able to detect when certain types of error occur in this circuit, so that we can discard erroneous measurement results. In error detection terminology, this is also known as our payload circuit." + "### Step 1: Map classical inputs to a quantum problem\n", + "\n", + "The payload circuit is a shallow one-dimensional brickwork random Clifford circuit. Because the circuit is Clifford, it prepares a stabilizer state whose fidelity you can estimate directly from sampled stabilizer expectation values. Start with a shallow circuit so that the checks are easy to visualize in the next step." ] }, { "cell_type": "code", - "execution_count": null, - "id": "f8384a7d", + "execution_count": 3, + "id": "step1-code", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, - "execution_count": 2, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "circ = random_clifford(num_qubits=2, seed=11).to_circuit()\n", - "circ.draw(\"mpl\")" - ] - }, - { - "cell_type": "markdown", - "id": "cf14cbd6", - "metadata": {}, - "source": [ - "### Step 2: Optimize problem for quantum hardware execution" + "num_qubits = 12\n", + "depth = 4\n", + "seed = 1764\n", + "rng = np.random.default_rng(seed)\n", + "np.random.seed(seed)\n", + "\n", + "circuit = random_clifford_circuit(num_qubits, depth, rng)\n", + "circuit.measure_all()\n", + "circuit.draw(\"mpl\", fold=-1, scale=0.6)" ] }, { "cell_type": "markdown", - "id": "aa41f799", + "id": "step2-header", "metadata": {}, "source": [ - "Our goal is to insert a coherent Pauli check into this payload circuit. But before we do that, we separate this circuit into layers. This will be useful later when inserting Pauli gates in between." + "### Step 2: Optimize problem for quantum hardware execution\n", + "\n", + "Mapping the circuit to hardware sets the physical qubit layout, the noise model that scores candidate checks, and the checks themselves.\n", + "\n", + "First, choose a one-dimensional qubit layout on the Heron r3 QPU `ibm_boston`. The layout zig-zags diagonally through the coupling graph to maximize the number of available ancilla qubits. The `get_check_qubits` function returns target and ancilla pairs, where a check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", + "\n", + "In the coupling graph that follows, the green qubits are payload qubits and the orange qubits are the ancillas that implement the checks. Qubits with an adjacent ancilla are used as target qubits for checks." ] }, { "cell_type": "code", - "execution_count": 3, - "id": "04bb6ea7", + "execution_count": 4, + "id": "step2-layout", "metadata": {}, "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Target qubits: [68, 69, 89, 91, 111, 113, 133]\n", + "Ancilla qubits: [67, 70, 88, 92, 110, 114, 132]\n" + ] + }, { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, + "execution_count": 4, "metadata": {}, - "output_type": "display_data" + "output_type": "execute_result" } ], "source": [ - "# Separate circuit into layers\n", - "dag = circuit_to_dag(circ)\n", - "circ_layers = []\n", - "for layer in dag.layers():\n", - " layer_as_circuit = dag_to_circuit(layer[\"graph\"])\n", - " circ_layers.append(layer_as_circuit)\n", - "\n", - "# Create subplots\n", - "fig, (ax1, ax2, ax3, ax4, ax5) = plt.subplots(1, 5, figsize=(10, 4))\n", - "\n", - "# Draw circuits on respective axes\n", - "circ_layers[0].draw(output=\"mpl\", ax=ax1)\n", - "circ_layers[1].draw(output=\"mpl\", ax=ax2)\n", - "circ_layers[2].draw(output=\"mpl\", ax=ax3)\n", - "circ_layers[3].draw(output=\"mpl\", ax=ax4)\n", - "circ_layers[4].draw(output=\"mpl\", ax=ax5)\n", - "\n", - "# Adjust layout to prevent overlap\n", - "plt.tight_layout()\n", - "plt.show()" + "service = QiskitRuntimeService()\n", + "backend = service.backend(\"ibm_marrakesh\")\n", + "\n", + "# Choose a layout and pair each available target qubit with a neighboring ancilla\n", + "layout = [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133]\n", + "target_qubits, ancilla_qubits = get_check_qubits(backend, layout)\n", + "num_checks = len(target_qubits)\n", + "\n", + "print(f\"Target qubits: {target_qubits}\")\n", + "print(f\"Ancilla qubits: {ancilla_qubits}\")\n", + "plot_coupling_map(\n", + " num_qubits=backend.num_qubits,\n", + " qubit_coordinates=getattr(\n", + " backend.configuration(), \"qubit_coordinates\", None\n", + " ),\n", + " coupling_map=backend.configuration().coupling_map,\n", + " figsize=(12, 12),\n", + " qubit_color=[\n", + " \"#4CAF50\"\n", + " if i in set(layout)\n", + " else \"#FF9800\"\n", + " if i in set(ancilla_qubits)\n", + " else \"#DDDDDD\"\n", + " for i in backend.coupling_map.graph.node_indices()\n", + " ],\n", + " qubit_size=220,\n", + " line_width=2,\n", + " font_size=90,\n", + ")" ] }, { "cell_type": "markdown", - "id": "dda1b961", - "metadata": {}, - "source": [ - "Now we are ready to add coherent Pauli checks into the payload circuit. To do this, we need to construct a \"valid check\" and insert it into the circuit. A \"check\" in this case is an operator that is capable of signaling whether an error occurred in the circuit by making a measurement on an ancilla qubit. It is considered a valid check when the additional operators inserted into the quantum circuit don't logically change the original circuit.\n", - "\n", - "This check is capable of detecting types of errors that anticommute with it, and the check will trigger a measurement of $\\ket{1}$ state in the ancilla qubit instead of $\\ket{0}$ through phase kickback. Therefore, we will be able to discard measurements where an error was signaled.\n", - "\n", - "In general, coherent Pauli checks are controlled-Pauli operators inserted into \"wires\" - spacetime locations between gates. The ancilla qubit responsible for signaling the error is the control qubit.\n", - "\n", - "Below we construct a valid check for the Clifford circuit we created above. We can demonstrate that this check doesn't change the circuit operation by showing that when these Pauli checks are propagated to the front of the circuit, they cancel each other out. This is easily shown because a Pauli operator through a Clifford gate is another Pauli operator.\n", - "\n", - "In general, one can use a decoding heuristic as outlined in [[1]](https://arxiv.org/abs/2504.15725) to identify valid checks. For the purposes of our initial example, we can also construct valid checks by using analytical Pauli and Clifford gate multiplication conditions." - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "id": "ca5727c1", + "id": "step2-transpile-md", "metadata": {}, - "outputs": [], "source": [ - "# Define a valid check\n", - "pauli_1 = Pauli(\"ZI\")\n", - "pauli_2 = Pauli(\"XZ\")" + "With the backend and layout chosen, transpile the payload into an instruction set architecture (ISA) circuit. Setting the layout and translating the gates into the native gate set of `ibm_boston` is enough here." ] }, { "cell_type": "code", "execution_count": 5, - "id": "e1c1e16d", + "id": "step2-transpile", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "execution_count": 5, @@ -254,1275 +358,646 @@ } ], "source": [ - "circ_1 = circ_layers[0].compose(circ_layers[1])\n", - "circ_1.draw(\"mpl\")" + "pm = generate_preset_pass_manager(\n", + " optimization_level=0, backend=backend, initial_layout=layout\n", + ")\n", + "circuit_isa = pm.run(circuit)\n", + "circuit_isa.draw(\"mpl\", fold=-1, scale=0.6)" ] }, { - "cell_type": "code", - "execution_count": 6, - "id": "bebdd7df", + "cell_type": "markdown", + "id": "step2-noise-md", "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "Pauli('-ZI')" - ] - }, - "execution_count": 6, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ - "pauli_1_ev = pauli_1.evolve(circ_1, frame=\"h\")\n", - "pauli_1_ev" + "Next, model how the gate and readout noise on the backend affects execution. The noise model determines where in the circuit a check captures the most error. A more accurate model improves the detection, but it is usually not necessary to learn one by sampling the QPU. The model that follows infers a uniform depolarizing channel for gate and readout noise from `qiskit-ibm-runtime` benchmark data." ] }, { "cell_type": "code", - "execution_count": 7, - "id": "246a42b8", + "execution_count": 6, + "id": "step2-noise", "metadata": {}, "outputs": [ { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" + "name": "stdout", + "output_type": "stream", + "text": [ + "NoiseModel(gate_noise=0.19074977552838765, readout_noise=0.1016845703125, idling_noise=None)\n" + ] } ], "source": [ - "circ_2 = circ.copy()\n", - "circ_2.draw(\"mpl\")" + "noise_model = NoiseModel.from_backend(\n", + " backend, layout, uniform_gate_noise=True\n", + ")\n", + "print(noise_model)" ] }, { - "cell_type": "code", - "execution_count": 8, - "id": "415c9471", + "cell_type": "markdown", + "id": "step2-checks-md", "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "Pauli('-ZI')" - ] - }, - "execution_count": 8, - "metadata": {}, - "output_type": "execute_result" - } - ], "source": [ - "pauli_2_ev = pauli_2.evolve(circ_2, frame=\"h\")\n", - "pauli_2_ev" + "Now add checks to the circuit. The `add_pauli_checks` function takes the Clifford payload, the list of target qubits, and the noise model. The `ancilla_qubits` argument tells the function which physical ancilla to pair with each target. Checks are added in the order the target qubits appear, so the final layout of the checked circuit is `layout + ancilla_qubits`. To run an output circuit with fewer (`i`) checks, the final layout is `layout + ancilla_qubits[:i]`.\n", + "\n", + "The output of `add_pauli_checks` is a sequence of circuits with an increasing number of checks, from no checks up to one check on every target qubit. The visualization confirms that the checks use the specified target and ancilla pairs. For details on finding good checks, see Sections II to IV of the supplementary information in reference [[1]](#references)." ] }, { "cell_type": "code", - "execution_count": 9, - "id": "f7368e97", + "execution_count": 7, + "id": "step2-checks", "metadata": {}, "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Physical layout of payload and ancillas: [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133, 67, 70, 88, 92, 110, 114, 132]\n", + "Checked circuit:\n" + ] + }, { "data": { "text/plain": [ - "Pauli('II')" + "\"Output" ] }, - "execution_count": 9, + "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ - "pauli_1_ev.dot(pauli_2_ev)" + "checked = add_pauli_checks(\n", + " circuit_isa,\n", + " target_qubits,\n", + " noise_model,\n", + " ancilla_qubits=ancilla_qubits,\n", + " cost=\"gamma\",\n", + " method=\"windowed\",\n", + " seed=seed,\n", + ")\n", + "\n", + "print(f\"Physical layout of payload and ancillas: {layout + ancilla_qubits}\")\n", + "print(\"Checked circuit:\")\n", + "checked[-1].circuit.draw(\"mpl\", fold=-1, idle_wires=False)" ] }, { "cell_type": "markdown", - "id": "dba1bc80", + "id": "step3-header", "metadata": {}, "source": [ - "As we can see, we have a valid check, since the inserted Pauli operators simply have the same effect as an identity operator on the circuit. We can now insert these checks into the circuit with an ancilla qubit. This ancilla qubit, or the check qubit, starts in the $\\ket{+}$ state. It includes the controlled versions of the Pauli operations outlined above and is finally measured in the $X$ basis. This check qubit is now capable of capturing errors in the payload circuit without logically altering it. This is because certain types of noise in the payload circuit will modify the state of the check qubit, and it will be measured \"1\" instead of \"0\" in case such an error occurs." + "### Step 3: Execute using Qiskit primitives\n", + "\n", + "To make the effect of gate noise visible, increase the depth of the payload and sample a subset of its stabilizers. Each stabilizer is generally not qubit-wise commuting with the others, so a single set of checks is not valid for two different stabilizers. Rather than group stabilizers into commuting sets, find a good set of checks for each stabilizer independently. Sampling stabilizers uniformly at random gives an unbiased fidelity estimate.\n", + "\n", + "Build the deeper circuit and draw a random sample of its stabilizers." ] }, { "cell_type": "code", - "execution_count": 10, - "id": "52add690", + "execution_count": 8, + "id": "step3-stabilizers", "metadata": {}, "outputs": [ { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" + "name": "stdout", + "output_type": "stream", + "text": [ + "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {-XYXZIZXYIIXZ, -XZYYIIXZXYYY, ...}\n" + ] } ], "source": [ - "# New circuit with 3 qubits (2 payload + 1 ancilla for check)\n", - "circ_meas = QuantumCircuit(3)\n", - "circ_meas.h(0)\n", - "circ_meas.compose(circ_layers[0], [1, 2], inplace=True)\n", - "circ_meas.compose(circ_layers[1], [1, 2], inplace=True)\n", - "circ_meas.cz(0, 2)\n", - "circ_meas.compose(circ_layers[2], [1, 2], inplace=True)\n", - "circ_meas.compose(circ_layers[3], [1, 2], inplace=True)\n", - "circ_meas.compose(circ_layers[4], [1, 2], inplace=True)\n", - "circ_meas.cz(0, 1)\n", - "circ_meas.cx(0, 2)\n", - "circ_meas.h(0)\n", - "\n", - "# Add measurement to payload qubits\n", - "c0 = ClassicalRegister(2, name=\"c0\")\n", - "circ_meas.add_register(c0)\n", - "circ_meas.measure(1, c0[0])\n", - "circ_meas.measure(2, c0[1])\n", - "\n", - "# Add measurement to check qubit\n", - "c1 = ClassicalRegister(1, name=\"c1\")\n", - "circ_meas.add_register(c1)\n", - "circ_meas.measure(0, c1[0])\n", - "\n", - "# Visualize the final circuit with the inserted checks\n", - "circ_meas.draw(\"mpl\")" - ] - }, - { - "cell_type": "markdown", - "id": "8ccacbe5", - "metadata": {}, - "source": [ - "### Step 3: Execute using Qiskit primitives" + "depth = 24\n", + "num_stabilizers = 20\n", + "num_shots = 1_000\n", + "\n", + "circuit = random_clifford_circuit(num_qubits, depth, rng)\n", + "\n", + "# Build the full stabilizer group, then sample from it uniformly at random\n", + "circ_no_meas = circuit.remove_final_measurements(inplace=False)\n", + "stabilizer_group = PauliList([Pauli(\"I\" * num_qubits)])\n", + "for generator in (\n", + " Pauli(label) for label in Clifford(circ_no_meas).to_labels(mode=\"S\")\n", + "):\n", + " stabilizer_group = stabilizer_group + stabilizer_group.compose(generator)\n", + "\n", + "keep = np.where(\n", + " stabilizer_group.x.any(axis=1) | stabilizer_group.z.any(axis=1)\n", + ")[0]\n", + "chosen = np.random.default_rng(seed).choice(\n", + " keep, size=min(num_stabilizers, len(keep)), replace=False\n", + ")\n", + "stabilizers = [stabilizer_group[int(i)] for i in chosen]\n", + "\n", + "two_qubit_depth = circuit.depth(lambda x: x.operation.num_qubits == 2)\n", + "print(\n", + " f\"Sampled {len(stabilizers)} stabilizers of a {circuit.num_qubits}-qubit \"\n", + " f\"circuit with two-qubit depth {two_qubit_depth}: \"\n", + " f\"{{{stabilizers[0]}, {stabilizers[1]}, ...}}\"\n", + ")" ] }, { "cell_type": "markdown", - "id": "ece387e6", + "id": "step3-findchecks-md", "metadata": {}, "source": [ - "If the check qubit is measured in \"0\", we keep that measurement. If it's measured in \"1\", then this means that an error occurred in the payload circuit, and we discard that measurement." + "For each sampled stabilizer, rotate the circuit so that the stabilizer is measured in the computational basis, transpile it onto the backend, and find a good set of checks. The target and ancilla pairs are shuffled together for each stabilizer so that each target keeps its ancilla. Remember that checks are committed sequentially in the order the target qubits are given, and a committed check is not changed as more checks are added." ] }, { "cell_type": "code", - "execution_count": 11, - "id": "78679785", + "execution_count": 9, + "id": "step3-findchecks", "metadata": {}, "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "100%|██████████| 20/20 [00:14<00:00, 1.37it/s]" + ] + }, { "name": "stdout", "output_type": "stream", "text": [ - "Stabilizer simulation result: {'0 11': 523, '0 01': 477}\n" + "Added 7 checks to 20 circuits in 15s.\n", + "On average, two-qubit depth increased from 24 to 33 when adding 7 checks.\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "\n" ] } ], "source": [ - "# Noiseless simulation using stabilizer method\n", - "sim_stab = AerSimulator(method=\"stabilizer\")\n", - "res = sim_stab.run(circ_meas, shots=1000).result()\n", - "counts_noiseless = res.get_counts()\n", - "print(f\"Stabilizer simulation result: {counts_noiseless}\")" + "noisy_circuits = []\n", + "checked_circuits = []\n", + "depths_2q = []\n", + "t0 = time.time()\n", + "for i, pauli in enumerate(tqdm(stabilizers)):\n", + " noisy_circuits.append(pm.run(append_basis_rotation(circuit, pauli)))\n", + " # Shuffle target and ancilla pairs together so each target keeps its ancilla\n", + " targets, ancillas = zip(\n", + " *random.sample(\n", + " list(zip(target_qubits, ancilla_qubits, strict=True)),\n", + " k=len(target_qubits),\n", + " ),\n", + " strict=True,\n", + " )\n", + " checked_circuits.append(\n", + " add_pauli_checks(\n", + " noisy_circuits[-1],\n", + " list(targets),\n", + " noise_model,\n", + " ancilla_qubits=list(ancillas),\n", + " cost=\"gamma\",\n", + " method=\"windowed\",\n", + " seed=seed + 1 + i,\n", + " )\n", + " )\n", + " depths_2q.append(\n", + " checked_circuits[-1][-1].circuit.depth(lambda x: len(x.qubits) == 2)\n", + " )\n", + "\n", + "print(\n", + " f\"Added {num_checks} checks to {len(stabilizers)} circuits \"\n", + " f\"in {(time.time() - t0):.0f}s.\"\n", + ")\n", + "print(\n", + " f\"On average, two-qubit depth increased from \"\n", + " f\"{circuit.depth(lambda x: len(x.qubits) == 2)} to {int(np.mean(depths_2q))} \"\n", + " f\"when adding {num_checks} checks.\"\n", + ")" ] }, { "cell_type": "markdown", - "id": "2bbbdbc4", + "id": "step3-sample-md", "metadata": {}, "source": [ - "Note that with an ideal simulator, the check qubit won't detect any errors as we will show in the subsequent post-processing section. We now introduce a noise model to the simulation and see how the check qubit captures errors." + "Sample the bare payload and the checked circuits with Qiskit Aer. The simulator uses the same depolarizing model that scored the checks, so the noise that the checks target is the noise that the simulator applies." ] }, { "cell_type": "code", - "execution_count": null, - "id": "407c2a8a", + "execution_count": 10, + "id": "step3-sample", "metadata": {}, "outputs": [ { - "name": "stdout", + "name": "stderr", "output_type": "stream", "text": [ - "Noise model simulation result: {'1 01': 5, '0 11': 478, '1 11': 6, '1 00': 2, '1 10': 1, '0 01': 500, '0 00': 5, '0 10': 3}\n" + "100%|██████████| 20/20 [00:19<00:00, 1.00it/s]\n" ] } ], "source": [ - "# Qiskit Aer noise model\n", - "noise = NoiseModel()\n", - "p2 = 0.003 # two-qubit depolarizing per CZ\n", - "p1 = 0.001 # one-qubit depolarizing per 1q Clifford\n", - "pr = 0.01 # readout bit-flip probability\n", - "\n", - "# 1q depolarizing on common 1q gates\n", - "e1 = depolarizing_error(p1, 1)\n", - "for g1 in [\"id\", \"rz\", \"sx\", \"x\", \"h\", \"s\"]:\n", - " noise.add_all_qubit_quantum_error(e1, g1)\n", - "\n", - "# 2q depolarizing on CZ\n", - "e2 = depolarizing_error(p2, 2)\n", - "noise.add_all_qubit_quantum_error(e2, \"cz\")\n", - "\n", - "# Readout error on measure\n", - "ro = ReadoutError([[1 - pr, pr], [pr, 1 - pr]])\n", - "noise.add_all_qubit_readout_error(ro)\n", - "\n", - "# Qiskit Aer simulation with noise model\n", - "aer = AerSimulator(method=\"automatic\", seed_simulator=43210)\n", - "job = aer.run(circ_meas, shots=1000, noise_model=noise)\n", - "result = job.result()\n", - "counts_noisy = result.get_counts()\n", - "print(f\"Noise model simulation result: {counts_noisy}\")" + "aer_nm = AerNoiseModel()\n", + "aer_nm.add_all_qubit_quantum_error(\n", + " depolarizing_error(noise_model.gate_noise, 2), [\"cz\"]\n", + ")\n", + "p = noise_model.readout_noise\n", + "aer_nm.add_all_qubit_readout_error(ReadoutError([[1 - p, p], [p, 1 - p]]))\n", + "noisy_sim = AerSimulator(method=\"stabilizer\", noise_model=aer_nm)\n", + "\n", + "counts = []\n", + "for i, checked_circ_result in enumerate(tqdm(checked_circuits)):\n", + " noisy_counts = (\n", + " noisy_sim.run(\n", + " noisy_circuits[i], shots=num_shots, seed_simulator=seed * i + 1\n", + " )\n", + " .result()\n", + " .get_counts()\n", + " )\n", + " checked_counts_per_variant = []\n", + " for k, ck in enumerate(checked_circ_result):\n", + " variant_counts = (\n", + " noisy_sim.run(\n", + " ck.circuit, shots=num_shots, seed_simulator=seed * i + 2 + k\n", + " )\n", + " .result()\n", + " .get_counts()\n", + " )\n", + " checked_counts_per_variant.append(variant_counts)\n", + " counts.append((noisy_counts, checked_counts_per_variant))" ] }, { "cell_type": "markdown", - "id": "800154ac", + "id": "step4-header", "metadata": {}, "source": [ "### Step 4: Post-process and return result in desired classical format\n", - "We can now retrieve and analyze the results from the Sampler job. We start by plotting results from the noiseless simulation." + "\n", + "Each check uses entangling gates between one ancilla and one target. The ancilla starts in $|0\\rangle$, so $Z_\\text{anc}$ stabilizes its input. Propagating $Z_\\text{anc}$ forward through the checked circuit yields a Pauli operator on the output whose non-identity terms define the check's support. A check passes when the bits in its support have even parity. A sample is kept only when every check passes.\n", + "\n", + "The `get_postselection_method` of each `CheckedCircuit` returns a function that maps a measured bitstring to a syndrome vector. Keep the samples whose syndrome is zero for every check, and discard the rest. The chart that follows shows that adding more checks lowers the postselection rate. A lower postselection rate needs more shots to reach a target accuracy, so there is a tradeoff between detection capability and sampling cost. The rate appears to converge, which indicates that additional checks contribute less detection capability." ] }, { "cell_type": "code", - "execution_count": 12, - "id": "4741910b", + "execution_count": 11, + "id": "step4-psr", "metadata": {}, "outputs": [ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, - "execution_count": 12, "metadata": {}, - "output_type": "execute_result" + "output_type": "display_data" } ], "source": [ - "# Plot the noiseless results\n", - "# Note that the first bit in the key corresponds to the check qubit\n", - "plot_histogram(counts_noiseless)" + "rate_per_variant = []\n", + "kept_per_stab = []\n", + "for i, (_, checked_counts_per_variant) in enumerate(counts):\n", + " rates = []\n", + " kept_at_num_checks = None\n", + " for k, variant_counts in enumerate(checked_counts_per_variant):\n", + " ps_fn = checked_circuits[i][k].get_postselection_method()\n", + " kept = {\n", + " bs: n for bs, n in variant_counts.items() if not ps_fn(bs).any()\n", + " }\n", + " rates.append(sum(kept.values()) / num_shots)\n", + " if k == num_checks:\n", + " kept_at_num_checks = kept\n", + " rate_per_variant.append(rates)\n", + " kept_per_stab.append(kept_at_num_checks)\n", + "\n", + "max_len = max(len(s) for s in rate_per_variant)\n", + "rates_arr = np.full((len(rate_per_variant), max_len), np.nan)\n", + "for i, s in enumerate(rate_per_variant):\n", + " rates_arr[i, : len(s)] = s\n", + "ks = np.arange(max_len)\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.plot(ks, rates_arr.T, color=\"#ff8c00\", alpha=0.15, linewidth=1)\n", + "ax.plot(\n", + " ks,\n", + " np.nanmedian(rates_arr, axis=0),\n", + " color=\"black\",\n", + " linewidth=1,\n", + " linestyle=\"--\",\n", + " label=\"median\",\n", + ")\n", + "ax.set_xlabel(\"Checks committed\")\n", + "ax.set_ylabel(\"Postselection rate\")\n", + "ax.set_ylim((0, 1.05))\n", + "ax.set_title(\n", + " f\"Per-stabilizer postselection rate ({len(rates_arr)} stabilizers)\"\n", + ")\n", + "ax.legend()\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()" ] }, { "cell_type": "markdown", - "id": "55d7c308", + "id": "step4-fidelity-md", "metadata": {}, "source": [ - "As expected, the check qubit doesn't detect any errors. Next, we plot results from the noisy simulation." + "Now compare the fidelity of the bare noisy state with the postselected state. Postselecting only the samples with no detected error raises the expectation value of every stabilizer, and therefore the estimated fidelity. The postselected values use fewer samples than the raw values, yet the expectation values are more accurate and the sampled variance is lower." ] }, { "cell_type": "code", - "execution_count": null, - "id": "51a8f3d9", + "execution_count": 12, + "id": "step4-fidelity", "metadata": {}, "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "ideal fidelity: 1.0\n", + "noisy fidelity: -0.0020\n", + "postselected fidelity: -0.0790\n", + "mean postselection rate: 0.007\n" + ] + }, { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, - "execution_count": 14, "metadata": {}, - "output_type": "execute_result" + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" } ], "source": [ - "# Plot results with noise model\n", - "plot_histogram(counts_noisy)" - ] - }, - { - "cell_type": "markdown", - "id": "e7d863b1", - "metadata": {}, - "source": [ - "As we can see, some measurements captured the error by flagging the check qubit as \"1\", which are visible in the last four columns. These shots are discarded." + "results = []\n", + "for i, ((noisy_counts, _), kept) in enumerate(\n", + " zip(counts, kept_per_stab, strict=True)\n", + "):\n", + " results.append(\n", + " (\n", + " expectation(noisy_counts, stabilizers[i]),\n", + " expectation(kept, stabilizers[i]),\n", + " sum(kept.values()) / num_shots,\n", + " )\n", + " )\n", + "\n", + "fidelity_noisy = float(np.nanmean([r[0] for r in results]))\n", + "fidelity_postsel = float(np.nanmean([r[1] for r in results]))\n", + "psr = float(np.mean([r[2] for r in results]))\n", + "print(\n", + " f\"ideal fidelity: 1.0\\n\"\n", + " f\"noisy fidelity: {fidelity_noisy:.4f}\\n\"\n", + " f\"postselected fidelity: {fidelity_postsel:.4f}\\n\"\n", + " f\"mean postselection rate: {psr:.3f}\"\n", + ")\n", + "\n", + "evs_ideal = np.ones(len(results))\n", + "evs_noisy = np.array([r[0] for r in results])\n", + "evs_post = np.array([r[1] for r in results])\n", + "idx = np.arange(len(results))\n", + "\n", + "\n", + "def strip(ax, ys, color, label):\n", + " m, s = np.nanmean(ys), np.nanstd(ys)\n", + " ax.axhspan(\n", + " m - s, m + s, color=color, alpha=0.15, label=f\"{label} mean and std\"\n", + " )\n", + " ax.axhline(m, color=color, linewidth=1, linestyle=\"--\")\n", + "\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.axhline(np.nanmean(evs_ideal), color=\"black\", linewidth=1.5, label=\"ideal\")\n", + "strip(ax, evs_noisy, \"red\", \"noisy\")\n", + "strip(ax, evs_post, \"green\", \"postselected\")\n", + "ax.scatter(idx, evs_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", + "ax.scatter(\n", + " idx,\n", + " evs_post,\n", + " color=\"green\",\n", + " s=22,\n", + " alpha=0.7,\n", + " label=\"postselected\",\n", + ")\n", + "ax.set_xlabel(\"stabilizer index\")\n", + "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", + "ax.set_ylim((-0.1, 1.1))\n", + "ax.set_title(\"Per-stabilizer expectation values\")\n", + "ax.legend(loc=\"lower left\")\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()\n", + "\n", + "M = np.arange(1, len(results) + 1)\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "for ys, color, label in [\n", + " (evs_ideal, \"black\", \"ideal\"),\n", + " (evs_noisy, \"red\", \"noisy\"),\n", + " (evs_post, \"green\", \"postselected\"),\n", + "]:\n", + " cm, sem = cum_mean_sem(ys)\n", + " ax.plot(M, cm, color=color, linewidth=1.5, label=label)\n", + " ax.fill_between(M, cm - sem, cm + sem, color=color, alpha=0.15)\n", + "ax.set_xlabel(\"number of stabilizers averaged\")\n", + "ax.set_ylabel(\"running fidelity estimate\")\n", + "ax.set_title(\"Fidelity convergence versus number of stabilizers\")\n", + "ax.legend()\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()" ] }, { "cell_type": "markdown", - "id": "dd4baf08", + "id": "step4-gamma-md", "metadata": {}, "source": [ - "Note: The ancilla qubit can also introduce new errors to the circuit. To reduce the effect of this, we can insert nested checks with additional ancilla qubits to the quantum circuit." + "The gamma score reports how much of the modeled noise channel remains undetected by the checks. Plotting the gamma score against the number of committed checks shows how the detection capability improves as each check is added. A gamma curve that converges toward `1.0` indicates that additional checks detect little extra error, which matches the convergence seen in the postselection rate." ] }, { - "cell_type": "markdown", - "id": "7027e13d", + "cell_type": "code", + "execution_count": 13, + "id": "step4-gamma", "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ - "## Large-scale hardware example" + "stab_scores = [\n", + " [variant.cost for variant in checked_circ_result]\n", + " for checked_circ_result in checked_circuits\n", + "]\n", + "max_len = max(len(s) for s in stab_scores)\n", + "scores = np.full((len(stab_scores), max_len), np.nan)\n", + "for i, s in enumerate(stab_scores):\n", + " scores[i, : len(s)] = s\n", + "ks = np.arange(max_len)\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.plot(ks, scores.T, color=\"#4682b4\", alpha=0.15, linewidth=1)\n", + "ax.plot(\n", + " ks,\n", + " np.nanmedian(scores, axis=0),\n", + " color=\"black\",\n", + " linewidth=1,\n", + " linestyle=\"--\",\n", + " label=\"median\",\n", + ")\n", + "ax.set_xlabel(\"Checks committed\")\n", + "ax.set_ylabel(\"Gamma\")\n", + "ax.set_yscale(\"log\")\n", + "ax.set_title(f\"Per-stabilizer gamma curves ({len(scores)} stabilizers)\")\n", + "ax.legend()\n", + "ax.grid(True, alpha=0.3, which=\"both\")\n", + "plt.tight_layout()\n", + "plt.show()" ] }, { "cell_type": "markdown", - "id": "c2b7f050", + "id": "hardware-header", "metadata": {}, "source": [ - "### Step 1: Map classical inputs to a quantum problem" + "## Large-scale hardware example\n", + "\n", + "The same workflow runs on hardware with a deeper payload and more stabilizers. This section reuses the backend, layout, target and ancilla pairs, noise model, and pass manager from the simulator example, then submits the sampled circuits to the QPU in a single job. Because the circuit is deeper, gate noise has a larger effect, and the fidelity gain from postselection is more pronounced.\n", + "\n", + "The parameters that follow set the depth, the number of stabilizers, and the number of shots. Increase `hw_num_stabilizers` for a more precise fidelity estimate at the cost of more circuits per job, and adjust the layout to use more qubits if you want a larger payload." ] }, { "cell_type": "markdown", - "id": "96ee4f90", + "id": "hardware-steps-header", "metadata": {}, "source": [ - "Now we demonstrate a significant task for quantum computing algorithms, which is preparing a GHZ state. We will demonstrate how to do this on a real backend using error detection." + "### Steps 1-4 (compressed into a single code block)\n", + "\n", + "The first cell builds the deeper payload, samples its stabilizers, finds the fully checked circuit for each one, and submits one Sampler job that contains both the bare and the checked circuits. Each job carries the tag `TUT_ASPC` so that you can find it later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." ] }, { "cell_type": "code", - "execution_count": null, - "id": "87317d7d", - "metadata": {}, - "outputs": [], - "source": [ - "# Set optional seed for reproducibility\n", - "SEED = 1\n", - "\n", - "if SEED:\n", - " np.random.seed(SEED)" - ] - }, - { - "cell_type": "markdown", - "id": "0eca54de", - "metadata": {}, - "source": [ - "The error detection algorithm for GHZ state preparation respects the hardware topology. We begin by selecting the desired hardware." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "78cafb7c", - "metadata": {}, - "outputs": [], - "source": [ - "# This is used to run on real hardware\n", - "service = QiskitRuntimeService()\n", - "\n", - "# Choose a backend to build GHZ on\n", - "backend_name = service.least_busy(\n", - " operational=True, simulator=False, min_num_qubits=133\n", - ")\n", - "\n", - "backend = service.backend(backend_name)\n", - "coupling_map = backend.target.build_coupling_map()" - ] - }, - { - "cell_type": "markdown", - "id": "12188865", - "metadata": {}, - "source": [ - "A GHZ state on $n$ qubits is defined as\n", - "$$\\lvert \\mathrm{GHZ}_n\\rangle \\;=\\; \\frac{1}{\\sqrt{2}}\\Big(\\lvert 0\\rangle^{\\otimes n} \\,+\\, \\lvert 1\\rangle^{\\otimes n}\\Big).$$\n", - "\n", - "A very naive approach to prepare the GHZ state would be to choose a root qubit with an initial Hadamard gate, which puts the qubit to an equal superposition state, and then to entangle this qubit with every other qubit. This is not a good approach, since it requires long-range and deep CNOT interactions. In this tutorial, we will use multiple techniques alongside error detection to reliably prepare the GHZ state on real hardware." - ] - }, - { - "cell_type": "markdown", - "id": "9dfcf756", - "metadata": {}, - "source": [ - "### Step 2: Optimize problem for quantum hardware execution" - ] - }, - { - "cell_type": "markdown", - "id": "cbb7cd7f", - "metadata": {}, - "source": [ - "#### Map the GHZ state to hardware" - ] - }, - { - "cell_type": "markdown", - "id": "181ce021", - "metadata": {}, - "source": [ - "First, we search for a root to map the GHZ circuit on hardware. We remove edges/nodes whose CZ errors, measurement errors, and $T_2$ values are worse than the thresholds below. These will not be included in the GHZ circuit." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "5033c96c", + "execution_count": 14, + "id": "hardware-submit", "metadata": {}, "outputs": [ { - "name": "stdout", + "name": "stderr", "output_type": "stream", "text": [ - "17 bad edges: \n", - "[[30, 31], [112, 113], [113, 114], [113, 119], [120, 121], [130, 131], [145, 146], [146, 147], [111, 112], [55, 59], [64, 65], [131, 138], [131, 132], [119, 133], [129, 130], [47, 57], [29, 38]]\n", - "5 bad nodes: \n", - "[1, 113, 131, 146, 120]\n" + "100%|██████████| 50/50 [03:34<00:00, 4.29s/it]\n" ] - } - ], - "source": [ - "def bad_cz(target, threshold=0.01):\n", - " \"\"\"Return list of edges whose CZ error is worse than threshold.\"\"\"\n", - " undirected_edges = []\n", - " for edge in backend.target.build_coupling_map().get_edges():\n", - " if (edge[1], edge[0]) not in undirected_edges:\n", - " undirected_edges.append(edge)\n", - " edges = undirected_edges\n", - " cz_errors = {}\n", - " for edge in edges:\n", - " cz_errors[edge] = target[\"cz\"][edge].error\n", - " worst_edges = sorted(cz_errors.items(), key=lambda x: x[1], reverse=True)\n", - " return [list(edge) for edge, error in worst_edges if error > threshold]\n", - "\n", - "\n", - "def bad_readout(target, threshold=0.01):\n", - " \"\"\"Return list of nodes whose measurement error is worse than threshold.\"\"\"\n", - " meas_errors = {}\n", - " for node in range(backend.num_qubits):\n", - " meas_errors[node] = target[\"measure\"][(node,)].error\n", - " worst_nodes = sorted(\n", - " meas_errors.items(), key=lambda x: x[1], reverse=True\n", - " )\n", - " return [node for node, error in worst_nodes if error > threshold]\n", - "\n", - "\n", - "def bad_coherence(target, threshold=60):\n", - " \"\"\"Return list of nodes whose T2 value is lower than threshold.\"\"\"\n", - " t2s = {}\n", - " for node in range(backend.num_qubits):\n", - " t2 = target.qubit_properties[node].t2\n", - " t2s[node] = t2 * 1e6 if t2 else 0\n", - " worst_nodes = sorted(t2s.items(), key=lambda x: x[1])\n", - " return [node for node, val in worst_nodes if val < threshold]\n", - "\n", - "\n", - "THRESH_CZ = 0.025 # exclude from BFS those edges whose\n", - "# CZ error is worse than this threshold\n", - "THRESH_MEAS = 0.15 # exclude from BFS those nodes whose\n", - "# measurement error is worse than this threshold\n", - "THRESH_T2 = 10 # exclude from BFS those nodes whose\n", - "# T2 value is lower than this threshold\n", - "\n", - "bad_edges = bad_cz(backend.target, threshold=THRESH_CZ)\n", - "bad_nodes_readout = bad_readout(backend.target, threshold=THRESH_MEAS)\n", - "dead_qubits = bad_readout(backend.target, threshold=0.4)\n", - "bad_nodes_coherence = bad_coherence(backend.target, threshold=THRESH_T2)\n", - "bad_nodes = list(set(bad_nodes_readout) | set(bad_nodes_coherence))\n", - "print(f\"{len(bad_edges)} bad edges: \\n{bad_edges}\")\n", - "print(f\"{len(bad_nodes)} bad nodes: \\n{bad_nodes}\")" - ] - }, - { - "cell_type": "markdown", - "id": "637c4584", - "metadata": {}, - "source": [ - "Using the function below, we construct the GHZ circuit on the chosen hardware starting from the root and using breadth-first search (BFS)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "5046c8ac", - "metadata": {}, - "outputs": [], - "source": [ - "def parallel_ghz(root, num_qubits, backend, bad_edges, skip):\n", - " \"\"\"\n", - " Build a GHZ state of size `num_qubits` on the given `backend`,\n", - " starting from `root`, expanding in BFS order.\n", - "\n", - " At each BFS layer, every active qubit adds at most one new neighbor\n", - " (so that two-qubit operations can run in parallel with no qubit conflicts).\n", - "\n", - " It grows the entanglement tree outward layer-by-layer.\n", - " \"\"\"\n", - "\n", - " # -------------------------------------------------------------\n", - " # (1) Filter usable connections from the backend coupling map\n", - " # -------------------------------------------------------------\n", - " # The coupling map lists all directed hardware connections as (control, target).\n", - " # We remove edges that are:\n", - " # - listed in `bad_edges` (or their reversed form)\n", - " # - involve a qubit in the `skip` list\n", - " cmap = [list(edge) for edge in backend.coupling_map.get_edges()]\n", - " edges = [\n", - " e\n", - " for e in cmap\n", - " if e not in bad_edges\n", - " and [e[1], e[0]] not in bad_edges\n", - " and e[0] not in skip\n", - " and e[1] not in skip\n", - " ]\n", - "\n", - " # -------------------------------------------------------------\n", - " # (2) Build an undirected adjacency list for traversal\n", - " # -------------------------------------------------------------\n", - " # Even though coupling_map edges are directed, BFS expansion just needs\n", - " # connectivity information (so we treat edges as undirected for search).\n", - " adj = defaultdict(list)\n", - " for u, v in edges:\n", - " adj[u].append(v)\n", - " adj[v].append(u)\n", - "\n", - " # -------------------------------------------------------------\n", - " # (3) Initialize the quantum circuit and BFS state\n", - " # -------------------------------------------------------------\n", - " n = backend.num_qubits\n", - " qc = QuantumCircuit(\n", - " n\n", - " ) # create a circuit with same number of qubits as hardware\n", - " visited = [\n", - " root\n", - " ] # record the order qubits are added to the GHZ chain/tree\n", - " queue = deque([root]) # BFS queue (start from root)\n", - " explored = defaultdict(\n", - " set\n", - " ) # to track which neighbors each node has already explored\n", - " layers = [] # list of per-layer (control, target) gate tuples\n", - " qc.h(root) # GHZ states start with a Hadamard on the root qubit\n", - "\n", - " # -------------------------------------------------------------\n", - " # (4) BFS expansion: build the GHZ tree one layer at a time\n", - " # -------------------------------------------------------------\n", - " # Loop until we've added the desired number of qubits to the GHZ\n", - " while queue and len(visited) < num_qubits:\n", - " layer = [] # collect new (control, target) pairs for this layer\n", - " current = list(\n", - " queue\n", - " ) # snapshot current frontier (so queue mutations don't affect iteration)\n", - " busy = (\n", - " set()\n", - " ) # track qubits already used in this layer (to avoid conflicts)\n", - "\n", - " for node in current:\n", - " queue.popleft()\n", - "\n", - " # find one unvisited neighbor of this node not already explored\n", - " unvisited_neighbors = [\n", - " nb\n", - " for nb in adj[node]\n", - " if nb not in visited and nb not in explored[node]\n", - " ]\n", - "\n", - " if unvisited_neighbors:\n", - " nb = unvisited_neighbors[\n", - " 0\n", - " ] # pick the first available neighbor\n", - " visited.append(nb) # mark it as part of the GHZ structure\n", - " queue.append(\n", - " node\n", - " ) # re-enqueue current node (can keep growing)\n", - " queue.append(nb) # enqueue the newly added qubit\n", - " explored[node].add(nb) # mark that edge as explored\n", - " layer.append(\n", - " (node, nb)\n", - " ) # schedule a CNOT between node and neighbor\n", - " busy.update([node, nb]) # reserve both qubits for this layer\n", - "\n", - " # stop early if we've reached the desired number of qubits\n", - " if len(visited) == num_qubits:\n", - " break\n", - " # else: node has no unused unvisited neighbors left → skip\n", - "\n", - " if layer:\n", - " # add all pairs (node, nb) scheduled this round to layers\n", - " layers.append(layer)\n", - " else:\n", - " # nothing new discovered this pass → done\n", - " break\n", - "\n", - " # -------------------------------------------------------------\n", - " # (5) Emit all layers into the quantum circuit\n", - " # -------------------------------------------------------------\n", - " # For each layer:\n", - " # - apply a CX gate for every (control, target) pair\n", - " # - insert a barrier so transpiler keeps layer structure\n", - " for layer in layers:\n", - " for q1, q2 in layer:\n", - " qc.cx(q1, q2)\n", - " qc.barrier()\n", - "\n", - " # -------------------------------------------------------------\n", - " # (6) Return outputs\n", - " # -------------------------------------------------------------\n", - " # qc: the built quantum circuit\n", - " # visited: order of qubits added\n", - " # layers: list of parallelizable two-qubit operations per step\n", - " return qc, visited, layers" - ] - }, - { - "cell_type": "markdown", - "id": "0167c0f2", - "metadata": {}, - "source": [ - "We now repeatedly search for the best root, where the GHZ circuit will originate from." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "90787353", - "metadata": {}, - "outputs": [], - "source": [ - "ROOT = None # root for BFS search\n", - "GHZ_SIZE = 100 # number of (data) qubits in the GHZ state\n", - "SKIP = [] # nodes to intentionally skip for a better chance of finding checks\n", - "\n", - "# Search for the best root (yielding the shallowest GHZ)\n", - "if ROOT is None:\n", - " best_root = -1\n", - " base_depth = 100\n", - " for root in range(backend.num_qubits):\n", - " qc, ghz_qubits, _ = parallel_ghz(\n", - " root, GHZ_SIZE, backend, bad_edges, SKIP\n", - " )\n", - " if len(ghz_qubits) != GHZ_SIZE:\n", - " continue\n", - " depth = qc.depth(lambda x: x.operation.num_qubits == 2)\n", - " if depth < base_depth:\n", - " best_root = root\n", - " base_depth = depth\n", - " ROOT = best_root" - ] - }, - { - "cell_type": "markdown", - "id": "9b8d8394", - "metadata": {}, - "source": [ - "We now construct the GHZ circuit starting from a specific node - as in, the best root - searching for the shortest depth using breadth-first search." - ] - }, - { - "cell_type": "code", - "execution_count": 20, - "id": "a04e921b", - "metadata": {}, - "outputs": [ + }, { "name": "stdout", "output_type": "stream", "text": [ - "base depth: 17, base count: 99\n", - "ROOT: 50\n" + "Submitted job d91ge2vqq29s738nkm00 with 100 circuits\n" ] } ], "source": [ - "# Build a GHZ starting at the best root\n", - "qc, ghz_qubits, _ = parallel_ghz(\n", - " ROOT, GHZ_SIZE, backend, bad_edges, SKIP + bad_nodes\n", + "# -------------------------Step 1: build a deeper payload and sample its stabilizers-------------------------\n", + "hw_depth = 72\n", + "hw_num_stabilizers = 50\n", + "hw_num_shots = 2_000\n", + "\n", + "hw_circuit = random_clifford_circuit(num_qubits, hw_depth, rng)\n", + "hw_no_meas = hw_circuit.remove_final_measurements(inplace=False)\n", + "hw_group = PauliList([Pauli(\"I\" * num_qubits)])\n", + "for generator in (\n", + " Pauli(label) for label in Clifford(hw_no_meas).to_labels(mode=\"S\")\n", + "):\n", + " hw_group = hw_group + hw_group.compose(generator)\n", + "keep = np.where(hw_group.x.any(axis=1) | hw_group.z.any(axis=1))[0]\n", + "chosen = np.random.default_rng(seed).choice(\n", + " keep, size=min(hw_num_stabilizers, len(keep)), replace=False\n", ")\n", - "base_depth = qc.depth(lambda x: x.operation.num_qubits == 2)\n", - "base_count = qc.size(lambda x: x.operation.num_qubits == 2)\n", - "print(f\"base depth: {base_depth}, base count: {base_count}\")\n", - "print(f\"ROOT: {ROOT}\")\n", - "if len(ghz_qubits) != GHZ_SIZE:\n", - " raise Exception(\"No GHZ found. Relax error thresholds.\")" - ] - }, - { - "cell_type": "markdown", - "id": "21bca6bf", - "metadata": {}, - "source": [ - "We need one final consideration before inserting valid checks. This is related to the concept of \"coverage\", which is a measure of how many of the wires in a quantum circuit a check can cover. With a higher coverage, we can detect errors on a wider part of the circuit. With this measure, we can select among the valid checks with the highest circuit coverage. In other words, we will be using the `weighted_coverage` function to score different checks for the GHZ circuit." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "a31f2dfb", - "metadata": {}, - "outputs": [], - "source": [ - "def weighted_coverage(layers, parities, w_idle=0.2, w_gate=0.8):\n", - " \"\"\"\n", - " Compute weighted fraction (idle + gate) of wires that are\n", - " covered by at least one parity to all active wires.\n", - " \"\"\"\n", - " wires = active_wires(layers) # defined below\n", - " covered_by_any = {n_layer: set() for n_layer in range(len(layers))}\n", - " for parity in parities:\n", - " trace = z_trace_backward(layers, parity) # defined below\n", - " for n_layer, qs in trace.items():\n", - " covered_by_any[n_layer] |= qs\n", - " covered_weight = 0\n", - " total_weight = 0\n", - " for n_layer in range(len(layers)):\n", - " idle = wires[n_layer][\"idle\"]\n", - " gate = wires[n_layer][\"gate\"]\n", - " total_weight += w_idle * len(idle) + w_gate * len(gate)\n", - " covered_idle = covered_by_any[n_layer] & idle\n", - " covered_gate = covered_by_any[n_layer] & gate\n", - " covered_weight += w_idle * len(covered_idle) + w_gate * len(\n", - " covered_gate\n", - " )\n", - " return covered_weight / total_weight if total_weight > 0 else 0\n", - "\n", - "\n", - "def active_wires(layers):\n", - " \"\"\"\n", - " Returns per-layer dict with two sets:\n", - " - 'idle': activated wires that are idle in this layer\n", - " - 'gate': activated wires that are control/target of a CNOT at this layer\n", - " \"\"\"\n", - " first_activation = {}\n", - " for n_layer, layer in enumerate(layers):\n", - " for c, t in layer:\n", - " first_activation.setdefault(c, n_layer)\n", - " first_activation.setdefault(t, n_layer)\n", - " result = {}\n", - " for n_layer in range(len(layers)):\n", - " active = {\n", - " q\n", - " for q, n_layer0 in first_activation.items()\n", - " if n_layer >= n_layer0\n", - " }\n", - " gate = {q for c, t in layers[n_layer] for q in (c, t)}\n", - " idle = active - gate\n", - " result[n_layer] = {\"idle\": idle, \"gate\": gate}\n", - " return result\n", - "\n", - "\n", - "def z_trace_backward(layers, initial_Zs):\n", - " \"\"\"\n", - " Backward propagate Zs with parity cancellation.\n", - " Returns {layer: set of qubits with odd parity Z at that layer}.\n", - " \"\"\"\n", - " wires = active_wires(layers)\n", - " support = set(initial_Zs)\n", - " trace = {}\n", - " for n_layer in range(len(layers) - 1, -1, -1):\n", - " active = wires[n_layer][\"idle\"] | wires[n_layer][\"gate\"]\n", - " trace[n_layer] = support & active\n", - " # propagate backwards\n", - " new_support = set()\n", - " for q in support:\n", - " hit = False\n", - " for c, t in layers[n_layer]:\n", - " if q == t: # Z on target: copy to control\n", - " new_support ^= {t, c} # toggle both\n", - " hit = True\n", - " break\n", - " elif q == c: # Z on control: passes through\n", - " new_support ^= {c}\n", - " hit = True\n", - " break\n", - " if not hit: # unaffected\n", - " new_support ^= {q}\n", - " support = new_support\n", - " return trace" - ] - }, - { - "cell_type": "markdown", - "id": "631f4797", - "metadata": {}, - "source": [ - "We can now insert checks to the GHZ circuit. Finding valid checks is very convenient for the GHZ state, since any two-qubit Pauli $Z$ operator $Z_i Z_j$ acting on any two qubits $i,j$ of the GHZ circuit is a support and therefore a valid check.\n", - "\n", - "Also note that the checks in this case are controlled-$Z$ operators neighboring Hadamard gates from the left and right on the ancilla qubit. This is equivalent to a CNOT gate applied to the ancilla qubit. The code below inserts the checks into the circuit." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "b7e798f7", - "metadata": {}, - "outputs": [], - "source": [ - "# --- Tunables controlling the search space / scoring ---\n", - "MAX_SKIPS = 10 # at most how many qubits to skip\n", - "# (in addition to the bad ones and the ones forced to skip above)\n", - "SHUFFLES = 200 # how many times to try removing nodes for checks\n", - "MAX_DEPTH_INCREASE = 10 # how far from the base GHZ depth to go\n", - "# to include checks (increase this for\n", - "# more checks at expense of depth)\n", - "\n", - "W_IDLE = 0.2 # weight of errors to consider during idle timesteps\n", - "W_GATE = 0.8 # weight of errors to consider during gates\n", - "\n", - "# Remove random nodes from the GHZ and build from the root\n", - "# again to increase checks\n", - "degree_two_nodes = [\n", - " i\n", - " for i in ghz_qubits\n", - " if all(n in ghz_qubits for n in coupling_map.neighbors(i))\n", - " and len(coupling_map.neighbors(i)) >= 2\n", - "]\n", - "\n", - "# --- Best-so-far tracking for the randomized search ---\n", - "num_checks = 0\n", - "best_covered_fraction = -1\n", - "best_qc = qc\n", - "best_checks = []\n", - "best_parities = []\n", - "best_layers = []\n", - "\n", - "# Outer loop: vary how many GHZ nodes we try skipping (0..MAX_SKIPS-1)\n", - "for num_skips in range(MAX_SKIPS):\n", - " # Inner loop: try SHUFFLES random choices of 'num_skips' nodes to skip\n", - " for _ in range(SHUFFLES):\n", - " # Construct the skip set:\n", - " # - pre-existing forced SKIP\n", - " # - plus a random sample of 'degree_two_nodes' of size 'num_skips'\n", - " skip = SKIP + list(np.random.choice(degree_two_nodes, num_skips))\n", - "\n", - " # Rebuild the GHZ using the current skip set and bad_nodes\n", - " qc, ghz_qubits, layers = parallel_ghz(\n", - " ROOT, GHZ_SIZE, backend, bad_edges, skip + bad_nodes\n", - " )\n", - "\n", - " # Measure circuit cost as 2-qubit-gate depth only\n", - " depth = qc.depth(lambda x: x.operation.num_qubits == 2)\n", - "\n", - " # If we failed to reach the target GHZ size, discard this attempt\n", - " if len(ghz_qubits) != GHZ_SIZE:\n", - " continue\n", - "\n", - " # --- Build \"checks\" around the GHZ we just constructed ---\n", - " # A check qubit is a non-GHZ, non-dead qubit that has ≥2\n", - " # neighbors inside the GHZ and all those incident\n", - " # edges are usable (i.e., not in bad_edges).\n", - " checks = []\n", - " parities = []\n", - " for i in range(backend.num_qubits):\n", - " neighbors = [\n", - " n for n in coupling_map.neighbors(i) if n in ghz_qubits\n", - " ]\n", - "\n", - " if (\n", - " i not in ghz_qubits\n", - " and i not in dead_qubits\n", - " and len(neighbors) >= 2\n", - " and not any(\n", - " [\n", - " [neighbor, i] in bad_edges\n", - " or [i, neighbor] in bad_edges\n", - " for neighbor in neighbors\n", - " ]\n", - " )\n", - " ):\n", - " # Record this qubit as a check qubit\n", - " checks.append(i)\n", - " parities.append((neighbors[0], neighbors[1]))\n", - " # Physically couple the check qubit 'i' to the two GHZ\n", - " # neighbors via CNOTs\n", - " # (This is the actual \"check\" attachment in the circuit.)\n", - " qc.cx(neighbors[0], i)\n", - " qc.cx(neighbors[1], i)\n", - "\n", - " # Score this design using the weighted coverage\n", - " # metric over the GHZ build layers\n", - " covered_fraction = weighted_coverage(\n", - " layers=layers, parities=parities, w_idle=W_IDLE, w_gate=W_GATE\n", - " )\n", + "hw_stabilizers = [hw_group[int(i)] for i in chosen]\n", + "\n", + "# -------------------------Step 2: transpile and add the fully checked circuit per stabilizer-------------------------\n", + "hw_noisy_circuits = []\n", + "hw_checked_circuits = []\n", + "for i, pauli in enumerate(tqdm(hw_stabilizers)):\n", + " bare = pm.run(append_basis_rotation(hw_circuit, pauli))\n", + " hw_noisy_circuits.append(bare)\n", + " variants = add_pauli_checks(\n", + " bare,\n", + " target_qubits,\n", + " noise_model,\n", + " ancilla_qubits=ancilla_qubits,\n", + " cost=\"gamma\",\n", + " method=\"windowed\",\n", + " seed=seed + 1 + i,\n", + " )\n", + " hw_checked_circuits.append(variants[-1]) # keep the fully checked circuit\n", "\n", - " # Keep it only if:\n", - " # - coverage improves over the best so far, AND\n", - " # - the 2q depth budget isn't blown by more than MAX_DEPTH_INCREASE\n", - " if (\n", - " covered_fraction > best_covered_fraction\n", - " and depth <= base_depth + MAX_DEPTH_INCREASE\n", - " ):\n", - " best_covered_fraction = covered_fraction\n", - " best_qc = qc\n", - " best_ghz_qubits = ghz_qubits\n", - " best_checks = checks\n", - " best_parities = parities\n", - " best_layers = layers" - ] - }, - { - "cell_type": "markdown", - "id": "8da8af69", - "metadata": {}, - "source": [ - "We can now print out the qubits used in the GHZ circuit and the check qubits." - ] - }, - { - "cell_type": "code", - "execution_count": 23, - "id": "e11e2392", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "GHZ qubits: [50, 49, 51, 38, 52, 48, 58, 53, 47, 71, 39, 46, 70, 54, 33, 45, 72, 69, 55, 32, 37, 73, 68, 34, 31, 44, 25, 74, 78, 67, 18, 24, 79, 75, 89, 57, 11, 23, 93, 59, 88, 66, 10, 22, 92, 90, 87, 65, 12, 9, 21, 94, 91, 86, 77, 13, 8, 20, 95, 98, 97, 14, 7, 36, 99, 111, 107, 15, 6, 41, 115, 110, 106, 19, 17, 5, 40, 114, 109, 108, 105, 27, 4, 42, 118, 104, 28, 3, 129, 117, 103, 29, 2, 128, 125, 96, 30, 127, 124, 102] 100\n", - "Check qubits: [16, 26, 35, 43, 85, 126] 6\n", - "Covered fraction (no idle): 0.4595959595959596\n" - ] - } - ], - "source": [ - "# --- After search, report the best design found ---\n", - "qc = best_qc\n", - "checks = best_checks\n", - "parities = best_parities\n", - "layers = best_layers\n", - "ghz_qubits = best_ghz_qubits\n", - "if len(ghz_qubits) != GHZ_SIZE:\n", - " raise Exception(\"No GHZ found. Relax error thresholds.\")\n", - "\n", - "print(f\"GHZ qubits: {ghz_qubits} {len(ghz_qubits)}\")\n", - "print(f\"Check qubits: {checks} {len(checks)}\")\n", - "\n", - "covered_fraction = weighted_coverage(\n", - " layers=layers, parities=parities, w_idle=W_IDLE, w_gate=W_GATE\n", - ")\n", - "print(\n", - " \"Covered fraction (no idle): \",\n", - " weighted_coverage(\n", - " layers=layers, parities=parities, w_idle=0.0, w_gate=1.0\n", - " ),\n", - ")" - ] - }, - { - "cell_type": "markdown", - "id": "4b0e4e74", - "metadata": {}, - "source": [ - "We can also print out some error statistics." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "040d4a25", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "cz errors: \n", - " mean: 0.002, max: 0.012\n", - "meas errors: \n", - " mean: 0.014, max: 0.121\n", - "t1 errors: \n", - " mean: 267.9, min: 23.6\n", - "t2 errors: \n", - " mean: 155.9, min: 13.9\n" - ] - } - ], - "source": [ - "def circuit_errors(target, circ, error_type=\"cz\"):\n", - " \"\"\"\n", - " Pull per-resource error numbers from a Qiskit Target\n", - " for ONLY the qubits/edges actually used by `circ`.\n", - "\n", - " Args:\n", - " target: qiskit.transpiler.Target (e.g., backend.target)\n", - " circ: qiskit.QuantumCircuit\n", - " error_type: one of {\"cz\", \"meas\", \"t1\", \"t2\"}:\n", - " - \"cz\" -> 2q CZ gate error on the circuit's used edges\n", - " - \"meas\" -> measurement error on the circuit's used qubits\n", - " - \"t1\" -> T1 (converted to microseconds) on used qubits\n", - " - \"t2\" -> T2 (converted to microseconds) on used qubits\n", - "\n", - " Returns:\n", - " list[float] of the requested quantity for the active edges/qubits.\n", - " \"\"\"\n", + "# -------------------------Step 3: submit one Sampler job with the bare and checked circuits-------------------------\n", + "sampler = Sampler(mode=backend)\n", + "sampler.options.default_shots = hw_num_shots\n", + "sampler.options.environment.job_tags = [\"TUT_ASPC\"]\n", "\n", - " # Get all 2-qubit edges that appear in the circuit (as undirected pairs).\n", - " active_edges = active_gates(circ) # e.g., {(0,1), (2,3), ...}\n", - "\n", - " # Intersect those with the device coupling map (so we only query valid edges).\n", - " # Note: target.build_coupling_map().get_edges() yields directed pairs.\n", - " edges = [\n", - " edge\n", - " for edge in target.build_coupling_map().get_edges()\n", - " if tuple(sorted(edge)) in active_edges\n", - " ]\n", - "\n", - " # Deduplicate direction: keep only one orientation of each edge.\n", - " undirected_edges = []\n", - " for edge in edges:\n", - " if (edge[1], edge[0]) not in undirected_edges:\n", - " undirected_edges.append(edge)\n", - " edges = undirected_edges # (not used later—see note below)\n", - "\n", - " # Accumulators for different error/physics quantities\n", - " cz_errors, meas_errors, t1_errors, t2_errors = [], [], [], []\n", - "\n", - " # For every active (undirected) edge in the circuit, fetch its CZ error.\n", - " # NOTE: Uses active_gates(circ) again (undirected tuples). This assumes\n", - " # `target['cz']` accepts undirected indexing;\n", - " # many Targets store both directions.\n", - " for edge in active_gates(circ):\n", - " cz_errors.append(target[\"cz\"][edge].error)\n", - "\n", - " # For every active qubit, fetch measure error and T1/T2 (converted to µs).\n", - " for qubit in active_qubits(circ):\n", - " meas_errors.append(target[\"measure\"][(qubit,)].error)\n", - " t1_errors.append(\n", - " target.qubit_properties[qubit].t1 * 1e6\n", - " ) # seconds -> microseconds\n", - " t2_errors.append(\n", - " target.qubit_properties[qubit].t2 * 1e6\n", - " ) # seconds -> microseconds\n", - "\n", - " # Select which set to return.\n", - " if error_type == \"cz\":\n", - " return cz_errors\n", - " elif error_type == \"meas\":\n", - " return meas_errors\n", - " elif error_type == \"t1\":\n", - " return t1_errors\n", - " else:\n", - " return t2_errors\n", - "\n", - "\n", - "def active_qubits(circ):\n", - " \"\"\"\n", - " Return a list of qubit indices that participate in at least one\n", - " non-delay, non-barrier instruction in `circ`.\n", - " \"\"\"\n", - " active_qubits = set()\n", - " for inst in circ.data:\n", - " # Skip scheduling artifacts that don't act on state\n", - " if (\n", - " inst.operation.name != \"delay\"\n", - " and inst.operation.name != \"barrier\"\n", - " ):\n", - " for qubit in inst.qubits:\n", - " q = circ.find_bit(\n", - " qubit\n", - " ).index # map Qubit object -> integer index\n", - " active_qubits.add(q)\n", - " return list(active_qubits)\n", - "\n", - "\n", - "def active_gates(circ):\n", - " \"\"\"\n", - " Return a set of undirected 2-qubit edges (i, j) that appear in `circ`.\n", - " \"\"\"\n", - " used_2q_gates = set()\n", - " for inst in circ:\n", - " if inst.operation.num_qubits == 2:\n", - " qs = inst.qubits\n", - " # map Qubit objects -> indices, then sort to make the edge undirected\n", - " qs = sorted([circ.find_bit(q).index for q in qs])\n", - " used_2q_gates.add(tuple(sorted(qs)))\n", - " return used_2q_gates\n", - "\n", - "\n", - "# ---- Print summary statistics ----\n", - "cz_errors = circuit_errors(backend.target, qc, error_type=\"cz\")\n", - "meas_errors = circuit_errors(backend.target, qc, error_type=\"meas\")\n", - "t1_errors = circuit_errors(backend.target, qc, error_type=\"t1\")\n", - "t2_errors = circuit_errors(backend.target, qc, error_type=\"t2\")\n", - "\n", - "np.set_printoptions(linewidth=np.inf)\n", - "print(\n", - " f\"cz errors: \\n mean: {np.round(np.mean(cz_errors), 3)}, \"\n", - " f\"max: {np.round(np.max(cz_errors), 3)}\"\n", - ")\n", - "print(\n", - " f\"meas errors: \\n mean: {np.round(np.mean(meas_errors), 3)}, \"\n", - " f\"max: {np.round(np.max(meas_errors), 3)}\"\n", - ")\n", - "print(\n", - " f\"t1 errors: \\n mean: {np.round(np.mean(t1_errors), 1)}, \"\n", - " f\"min: {np.round(np.min(t1_errors), 1)}\"\n", - ")\n", - "print(\n", - " f\"t2 errors: \\n mean: {np.round(np.mean(t2_errors), 1)}, \"\n", - " f\"min: {np.round(np.min(t2_errors), 1)}\"\n", - ")" + "pubs = hw_noisy_circuits + [cc.circuit for cc in hw_checked_circuits]\n", + "job = sampler.run(pubs)\n", + "print(f\"Submitted job {job.job_id()} with {len(pubs)} circuits\")" ] }, { "cell_type": "markdown", - "id": "a8f0a54f", + "id": "hardware-retrieve-md", "metadata": {}, "source": [ - "As previously, we can simulate the circuit first in the absence of noise to ensure correctness of the GHZ state preparation circuit." + "After the job completes, retrieve the results and postselect the checked counts. Compute the noisy and postselected fidelities as the average stabilizer expectation value over the sampled stabilizers." ] }, { "cell_type": "code", - "execution_count": null, - "id": "b905e875", + "execution_count": 15, + "id": "hardware-retrieve", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "Stabilizer simulation result:\n", - "{'000000 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111': 525, '000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000': 475}\n" + "noisy fidelity: 0.0032\n", + "postselected fidelity: 0.0268\n", + "mean postselection rate: 0.008\n" ] }, { "data": { "text/plain": [ - "\"Output" - ] - }, - "execution_count": 23, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# --- Simulate to ensure correctness ---\n", - "\n", - "qc_meas = qc.copy()\n", - "\n", - "# Add measurements to the GHZ qubits\n", - "c1 = ClassicalRegister(len(ghz_qubits), \"c1\")\n", - "qc_meas.add_register(c1)\n", - "for q, c in zip(ghz_qubits, c1):\n", - " qc_meas.measure(q, c)\n", - "\n", - "# Add measurements to the check qubits\n", - "if len(checks) > 0:\n", - " c2 = ClassicalRegister(len(checks), \"c2\")\n", - " qc_meas.add_register(c2)\n", - " for q, c in zip(checks, c2):\n", - " qc_meas.measure(q, c)\n", - "\n", - "# Simulate the circuit with stabilizer method\n", - "sim_stab = AerSimulator(method=\"stabilizer\")\n", - "res = sim_stab.run(qc_meas, shots=1000).result()\n", - "counts = res.get_counts()\n", - "print(\"Stabilizer simulation result:\")\n", - "print(counts)\n", - "\n", - "# Rename keys to \"0 0\" and \"0 1\" for easier plotting\n", - "# First len(checks) bits are check bits, rest are GHZ bits\n", - "keys = list(counts.keys())\n", - "for key in keys:\n", - " check_bits = key[: len(checks)]\n", - " ghz_bits = key[(len(checks) + 1) :]\n", - " if set(check_bits) == {\"0\"} and set(ghz_bits) == {\"0\"}:\n", - " counts[\"0 0\"] = counts.pop(key)\n", - " elif set(check_bits) == {\"0\"} and set(ghz_bits) == {\"1\"}:\n", - " counts[\"0 1\"] = counts.pop(key)\n", - " else:\n", - " continue\n", - "\n", - "plot_histogram(counts)" - ] - }, - { - "cell_type": "markdown", - "id": "a5db4e3b", - "metadata": {}, - "source": [ - "As expected, the check qubits are measured as all zeros, and we successfully prepared the GHZ state." - ] - }, - { - "cell_type": "markdown", - "id": "14770785", - "metadata": {}, - "source": [ - "### Step 3: Execute using Qiskit primitives" - ] - }, - { - "cell_type": "markdown", - "id": "d7a6e8a8", - "metadata": {}, - "source": [ - "Now we are ready to run the circuit on real hardware and demonstrate how the error detection protocol can capture errors in GHZ state preparation." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "e2d37555", - "metadata": {}, - "outputs": [], - "source": [ - "SHOTS = 10000 # number of shots" - ] - }, - { - "cell_type": "markdown", - "id": "eef98bd0", - "metadata": {}, - "source": [ - "We define a helper function to add measurements to the GHZ circuit." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "5f99ba1f", - "metadata": {}, - "outputs": [], - "source": [ - "def add_measurements(qc, ghz_qubits, checks):\n", - " # --- Measure each set of qubits into different\n", - " # classical registers to facilitate post-processing ---\n", - "\n", - " # Add measurements to the GHZ qubits\n", - " c1 = ClassicalRegister(len(ghz_qubits), \"c1\")\n", - " qc.add_register(c1)\n", - " for q, c in zip(ghz_qubits, c1):\n", - " qc.measure(q, c)\n", - "\n", - " # Add measurements to the check qubits\n", - " c2 = ClassicalRegister(len(checks), \"c2\")\n", - " qc.add_register(c2)\n", - " for q, c in zip(checks, c2):\n", - " qc.measure(q, c)\n", - "\n", - " return qc" - ] - }, - { - "cell_type": "markdown", - "id": "19eae8b4", - "metadata": {}, - "source": [ - "Before execution, we draw the layout of the GHZ qubits and the check qubits on the selected hardware." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "d0faf114", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" + "\"Output" ] }, "metadata": {}, @@ -1530,348 +1005,85 @@ } ], "source": [ - "# Plot the layout of GHZ and check qubits on the device\n", - "plot_gate_map(\n", - " backend,\n", - " label_qubits=True,\n", - " line_width=20,\n", - " line_color=[\n", - " \"black\"\n", - " if edge[0] in ghz_qubits + checks and edge[1] in ghz_qubits + checks\n", - " else \"lightgrey\"\n", - " for edge in backend.coupling_map.graph.edge_list()\n", - " ],\n", - " qubit_color=[\n", - " \"blue\"\n", - " if i in ghz_qubits\n", - " else \"salmon\"\n", - " if i in checks\n", - " else \"lightgrey\"\n", - " for i in range(0, backend.num_qubits)\n", - " ],\n", - ")\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": 29, - "id": "78c8c2b6", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 27, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "qc.draw(\"mpl\", idle_wires=False, fold=-1)" - ] - }, - { - "cell_type": "markdown", - "id": "681d4f29", - "metadata": {}, - "source": [ - "We now add the measurements." - ] - }, - { - "cell_type": "code", - "execution_count": 30, - "id": "75169ff2", - "metadata": {}, - "outputs": [], - "source": [ - "qc = add_measurements(qc, ghz_qubits, checks)" - ] - }, - { - "cell_type": "markdown", - "id": "9bf787f5", - "metadata": {}, - "source": [ - "The scheduling pipeline below locks in timing, removes barriers, simplifies delays, and injects dynamical decoupling, all while preserving the original operation times." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "673d1a87", - "metadata": {}, - "outputs": [], - "source": [ - "# The scheduling consists of first inserting delays while barriers\n", - "# are still there, then removing the barriers and consolidating the\n", - "# delays, so that the operations do not move in time\n", - "# Lastly we replace delays with dynamical decoupling\n", - "collect_function = partial(\n", - " collect_using_filter_function,\n", - " filter_function=(lambda node: node.op.name == \"delay\"),\n", - " split_blocks=True,\n", - " min_block_size=2,\n", - " split_layers=False,\n", - " collect_from_back=False,\n", - " max_block_width=None,\n", - ")\n", - "\n", - "collapse_function = partial(\n", - " collapse_to_operation,\n", - " collapse_function=(\n", - " lambda circ: Delay(sum(inst.operation.duration for inst in circ))\n", - " ),\n", - ")\n", - "\n", - "\n", - "class Unschedule(AnalysisPass):\n", - " \"\"\"Removes a property from the passmanager property set so that the\n", - " circuit looks unscheduled, so we can schedule it again.\"\"\"\n", - "\n", - " def run(self, dag):\n", - " del self.property_set[\"node_start_time\"]\n", - "\n", - "\n", - "def build_passmanager(backend, dd_qubits=None):\n", - " pm = generate_preset_pass_manager(\n", - " target=backend.target,\n", - " layout_method=\"trivial\",\n", - " optimization_level=2,\n", - " routing_method=\"none\",\n", - " )\n", - "\n", - " pm.scheduling = PassManager(\n", - " [\n", - " ALAPScheduleAnalysis(target=backend.target),\n", - " PadDelay(target=backend.target),\n", - " RemoveBarriers(),\n", - " Unschedule(),\n", - " CollectAndCollapse(\n", - " collect_function=collect_function,\n", - " collapse_function=collapse_function,\n", - " ),\n", - " ALAPScheduleAnalysis(target=backend.target),\n", - " PadDynamicalDecoupling(\n", - " dd_sequence=[XGate(), RZGate(-np.pi), XGate(), RZGate(np.pi)],\n", - " spacing=[1 / 4, 1 / 2, 0, 0, 1 / 4],\n", - " target=backend.target,\n", - " qubits=dd_qubits,\n", - " ),\n", - " ]\n", + "# -------------------------Step 4: postselect and compare fidelity-------------------------\n", + "result = job.result()\n", + "n_stab = len(hw_stabilizers)\n", + "\n", + "hw_results = []\n", + "for i in range(n_stab):\n", + " noisy_counts = result[i].join_data().get_counts()\n", + " checked_counts = result[n_stab + i].join_data().get_counts()\n", + " ps_fn = hw_checked_circuits[i].get_postselection_method()\n", + " kept = {bs: c for bs, c in checked_counts.items() if not ps_fn(bs).any()}\n", + " hw_results.append(\n", + " (\n", + " expectation(noisy_counts, hw_stabilizers[i]),\n", + " expectation(kept, hw_stabilizers[i]),\n", + " sum(kept.values()) / sum(checked_counts.values()),\n", + " )\n", " )\n", "\n", - " return pm" - ] - }, - { - "cell_type": "markdown", - "id": "6b6c4078", - "metadata": {}, - "source": [ - "We can now use the custom pass manager to transpile the circuit for the selected backend." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "8b8c537b", - "metadata": {}, - "outputs": [], - "source": [ - "# Transpile the circuits for the backend\n", - "pm = build_passmanager(backend, ghz_qubits)\n", - "\n", - "# Instruction set architecture (ISA) level circuit after scheduling and\n", - "# DD insertion\n", - "isa_circuit = pm.run(qc)\n", - "\n", - "# Draw after scheduling and DD insertion\n", - "# timeline_drawer(isa_circuit, show_idle=False, time_range=(0, 1000),\n", - "# target=backend.target)" - ] - }, - { - "cell_type": "code", - "execution_count": 33, - "id": "0c4b279b", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 31, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "isa_circuit.draw(\"mpl\", fold=-1, idle_wires=False)" - ] - }, - { - "cell_type": "markdown", - "id": "4e0712a5", - "metadata": {}, - "source": [ - "We then submit the job using the Qiskit Runtime Sampler primitive." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "803e5790", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "d493f17nmdfs73abf9qg\n" - ] - } - ], - "source": [ - "# Select the Sampler options\n", - "sampler = Sampler(mode=backend)\n", - "sampler.options.default_shots = SHOTS\n", - "sampler.options.dynamical_decoupling.enable = False\n", - "sampler.options.execution.rep_delay = 0.00025\n", - "sampler.options.environment.job_tags = [\"TUT_EDSC\"]\n", - "\n", - "# Submit the job\n", - "print(\"Submitting Sampler job\")\n", - "ghz_job = sampler.run([isa_circuit])\n", - "\n", - "print(ghz_job.job_id())" - ] - }, - { - "cell_type": "markdown", - "id": "c39bfe38", - "metadata": {}, - "source": [ - "### Step 4: Post-process and return result in desired classical format\n", - "We can now retrieve and analyze the results from the Sampler job." - ] - }, - { - "cell_type": "code", - "execution_count": 33, - "id": "23495738", - "metadata": {}, - "outputs": [], - "source": [ - "# Retrieve the job results\n", - "job_result = ghz_job.result()" - ] - }, - { - "cell_type": "code", - "execution_count": 34, - "id": "55e07409", - "metadata": {}, - "outputs": [], - "source": [ - "# Get the counts from GHZ and check qubit measurements\n", - "ghz_counts = job_result[0].data.c1.get_counts()\n", - "checks_counts = job_result[0].data.c2.get_counts()" - ] - }, - { - "cell_type": "code", - "execution_count": 35, - "id": "d7162ed7", - "metadata": {}, - "outputs": [], - "source": [ - "# Post-process to get unflagged GHZ counts (i.e., check bits are all '0')\n", - "joined_counts = job_result[0].join_data().get_counts()\n", - "unflagged_counts = {}\n", - "for key, count in joined_counts.items():\n", - " check_bits = key[: len(checks)]\n", - " ghz_bits = key[len(checks) :]\n", - " if set(check_bits) == {\"0\"}:\n", - " unflagged_counts[ghz_bits] = count" - ] - }, - { - "cell_type": "code", - "execution_count": 36, - "id": "d7902975", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "execution_count": 36, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Get top 20 outcomes by frequency from the unflagged counts\n", - "top_counts = dict(\n", - " sorted(unflagged_counts.items(), key=lambda x: x[1], reverse=True)[:20]\n", + "hw_fidelity_noisy = float(np.nanmean([r[0] for r in hw_results]))\n", + "hw_fidelity_postsel = float(np.nanmean([r[1] for r in hw_results]))\n", + "hw_psr = float(np.mean([r[2] for r in hw_results]))\n", + "print(\n", + " f\"noisy fidelity: {hw_fidelity_noisy:.4f}\\n\"\n", + " f\"postselected fidelity: {hw_fidelity_postsel:.4f}\\n\"\n", + " f\"mean postselection rate: {hw_psr:.3f}\"\n", ")\n", "\n", - "# Rename keys for better visualization\n", - "top_counts_renamed = {}\n", - "i = 0\n", - "for key, count in top_counts.items():\n", - " if set(key) == {\"0\"}:\n", - " top_counts_renamed[\"all 0s\"] = count\n", - " elif set(key) == {\"1\"}:\n", - " top_counts_renamed[\"all 1s\"] = count\n", - " else:\n", - " top_counts_renamed[f\"other_{i}\"] = count\n", - " i += 1\n", - "\n", - "plot_histogram(top_counts_renamed, figsize=(12, 7))" - ] - }, - { - "cell_type": "markdown", - "id": "90505f1c", - "metadata": {}, - "source": [ - "In the histogram above, we plotted 20 bitstring measurements from the GHZ qubits that were not flagged by the check qubits. As expected, all-0 and all-1 bitstrings have the highest counts. Note that some erroneous bitstrings with low error weights were not captured by the error detection. The highest counts are still found in the expected bitstrings." + "hw_noisy = np.array([r[0] for r in hw_results])\n", + "hw_post = np.array([r[1] for r in hw_results])\n", + "idx = np.arange(n_stab)\n", + "\n", + "fig, ax = plt.subplots(figsize=(8, 4))\n", + "ax.axhline(1.0, color=\"black\", linewidth=1.5, label=\"ideal\")\n", + "strip(ax, hw_noisy, \"red\", \"noisy\")\n", + "strip(ax, hw_post, \"green\", \"postselected\")\n", + "ax.scatter(idx, hw_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", + "ax.scatter(\n", + " idx,\n", + " hw_post,\n", + " color=\"green\",\n", + " s=22,\n", + " alpha=0.7,\n", + " label=\"postselected\",\n", + ")\n", + "ax.set_xlabel(\"stabilizer index\")\n", + "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", + "ax.set_ylim((-0.1, 1.1))\n", + "ax.set_title(\"Per-stabilizer expectation values on hardware\")\n", + "ax.legend(loc=\"lower left\")\n", + "ax.grid(True, alpha=0.3)\n", + "plt.tight_layout()\n", + "plt.show()" ] }, { "cell_type": "markdown", - "id": "5423c559", + "id": "next-steps", "metadata": {}, "source": [ "## Next steps\n", "\n", "\n", "If you found this work interesting, you might be interested in the following material:\n", - "- To further explore technical details of GHZ state preparation, see [[3]](https://arxiv.org/abs/2510.09520). In addition to error detection, authors use readout error mitigation with M3 and TREX and perform temporary uncomputation techniques to prepare high-fidelity GHZ states.\n", - "- Tutorial on [repetition codes](/docs/tutorials/repetition-codes).\n", - "- The tutorial [Automated spacetime Pauli checks with qiskit-paulice](/docs/tutorials/automated-spacetime-pauli-checks), which automates check finding with the `qiskit-paulice` package and boosts sampled Clifford-circuit fidelity through postselection.\n", + "\n", + "- The tutorial on [repetition codes](/docs/tutorials/repetition-codes) for an introduction to quantum error correction.\n", + "- The [`qiskit-paulice` package](https://github.com/Qiskit/qiskit-paulice) for the full check-finding API.\n", + "- The paper [Low-overhead error detection with spacetime codes](https://arxiv.org/abs/2504.15725) for the theory behind the checks.\n", "" ] }, { "cell_type": "markdown", - "id": "7bbdd161", + "id": "references", "metadata": {}, "source": [ "## References\n", - "- [1] Martiel, S., & Javadi-Abhari, A. (2025). Low-overhead error detection with spacetime codes. *arXiv preprint arXiv:2504.15725.*\n", - "- [2] van den Berg, E., Bravyi, S., Gambetta, J. M., Jurcevic, P., Maslov, D., & Temme, K. (2023). Single-shot error mitigation by coherent Pauli checks. *Physical Review Research*, 5(3), 033193.\n", - "- [3] Javadi-Abhari, A., Martiel, S., Seif, A., Takita, M., & Wei, K. X. (2025). Big cats: entanglement in 120 qubits and beyond. *arXiv preprint arXiv:2510.09520*." + "\n", + "- [1] Martiel, S., & Javadi-Abhari, A. (2025). Low-overhead error detection with spacetime codes. *arXiv preprint* [arXiv:2504.15725](https://arxiv.org/abs/2504.15725).\n", + "- [2] van den Berg, E., Bravyi, S., Gambetta, J. M., Jurcevic, P., Maslov, D., & Temme, K. (2023). Single-shot error mitigation by coherent Pauli checks. *Physical Review Research*, 5(3), 033193." ] } ], diff --git a/docs/tutorials/index.mdx b/docs/tutorials/index.mdx index 828605fb95c4..3ad267b2f46c 100644 --- a/docs/tutorials/index.mdx +++ b/docs/tutorials/index.mdx @@ -173,8 +173,6 @@ Error detection identifies faulty operations to return noise-free results shot-b * [Low-overhead error detection with spacetime codes](/docs/tutorials/ghz-spacetime-codes) -* [Automated spacetime Pauli checks with qiskit-paulice](/docs/tutorials/automated-spacetime-pauli-checks) - diff --git a/public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/hardware-retrieve-1.avif b/public/docs/images/tutorials/automated-spacetime-pauli-checks/extracted-outputs/hardware-retrieve-1.avif deleted file mode 100644 index 92d8e6c00e2e4182d5feecd1895557034dbda397..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 8560 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z+(r!1O8*k~FF}=|Nj%!J!^wZrnKIrHCw-)7BZ13XEq>z^;%gkMq@GizIyG4f17o%M zM8tv`K9AIk$u?FnTbL&Myu{&?(%GajTcOGyp^qXY7#+vwGut*Q$fP}&UkPX@*U7?M z<|B5p{he`6@_-{HV?IPs=1!p_Fk&n3v)IAe_F0vnX7ml~T4%k2TWjym#=6eu%;D44 zMk2cj^EyAYqXT62bT0+-)Cjz+u~;t~5O3;gQSyGEDaxj}IPgoVl%!5~xH{GH=2xwr z!WQ>9V$;a&56>u0KRmGG7tv-X*vJZgI3xxrKQM>t96I1YxsEmK_Kr6dXr?C9wk{?ypai zD;BBYl~{Ua9dD-$YEY5@i3lI{Sn*Y;jqW?V96(@zhhTXiFAI3{S;qP+u~}NTU0Y0QtD?na4h^b*V{<4 zQsR-nGoW#iH&o$Ue78@^X!LK%OWs0E*40xpyi@NQw$lU;dz>gS-evm zJQBE*jU2WIsA6@QC6i ze^|@T&*F(_g7K))SXD!? zwc?}W)4;A>iWJ^=!^G7^!Yfw)Ri8Tz_}&kU4bikzeji_2&43a7`hwrI@6=un^-#CM z?XXdwB}^7({7IcIG6XV)cUOf*|L}?ipMI^>j1Q!~RlzHoGA7rV7>SO~bzp4IPwXJN zRGTX2rswjn)ccCTBlKxO8sDEd^P1f3j<} zCFAd{sN3#OTuDR-<^Tl5)LCLkBC@{qL_Z4=QxO(%pmbvk@1G|g4!mLW)%Mx}V53)F4kGeV)SNP{RE zy21G#(@a0qCmC$tUFAwW4hhL3TJmHnL->-zo6YS|Ak3%dVy*MsNetNND^ErOY5Y_b z=-MUv6J5LS)hh1A^;O(>x(GdmZ&Y)zCKB%Lk|-2&AE{`SyOwy1R(vqMf#_PHKMJIT zFiZ4I*?+I#%nx@`4-k=-9Lmtl;=j&D=-LqL{ ziCAfrKV2r>DUO-G=Oc{lx%3GS8oO>gsck)wfLm69daDny_j;Yy6DRIP6?m6paR_k} z?}Am#$VfA#H6rF(b$TBZjB9Un6Z>oXNJyIq(B{dlu=AF*4Ep2zQDR|jKT8e6m4#~0 zTEul?Q0G$$DI2lNjMe+Se!nCA`+f)daWK3R2C)W$Ndd2mv`& Date: Wed, 1 Jul 2026 17:11:26 -0400 Subject: [PATCH 03/14] Update docs/tutorials/ghz-spacetime-codes.ipynb Co-authored-by: Rebecca Dimock <66339736+beckykd@users.noreply.github.com> --- docs/tutorials/ghz-spacetime-codes.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index fa4bfdb49d37..5ed09c2fb615 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -56,7 +56,7 @@ "\n", "### What the qiskit-paulice package does\n", "\n", - "The `qiskit-paulice` package automates the construction of these checks so that you do not have to build them by hand. Its primary role is to find and insert valid spacetime Pauli checks at the locations in a circuit that maximize error detection while minimizing qubit overhead. A check is *valid* when its operators leave the logical action of the payload circuit unchanged, *low-weight* when it uses few entangling gates, and *effective* when it detects a large fraction of the error relative to the noise the check itself introduces. The package scores candidate checks against a noise model and commits the best ones to the circuit. This tutorial uses three pieces of the API:\n", + "The `qiskit-paulice` package automates the construction of these checks so that you do not have to build them by hand. Its primary role is to find and insert valid spacetime Pauli checks at the locations in a circuit that maximize error detection while minimizing qubit overhead. A check is *valid* when its operators leave the logical action of the payload circuit unchanged, *low-weight* when it uses few entangling gates, and *effective* when it detects a large portion of the errors, relative to the noise the check itself introduces. The package scores candidate checks against a noise model and commits the best ones to the circuit. This tutorial uses three API methods:\n", "\n", "- `get_check_qubits` inspects a backend coupling map and returns target and ancilla qubit pairs. A check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", "- `NoiseModel.from_backend` builds a rough noise model from backend benchmark data. The model scores candidate checks, so an exact, learned noise model is not required. For a learned Pauli-Lindblad model, see `NoiseModel.from_pauli_lindblad_maps`.\n", From 9d8f42081e17b44f5e9c91f89eba6676f920e11e Mon Sep 17 00:00:00 2001 From: Henry Zou <87874865+henryzou50@users.noreply.github.com> Date: Wed, 1 Jul 2026 17:11:50 -0400 Subject: [PATCH 04/14] Update docs/tutorials/ghz-spacetime-codes.ipynb Co-authored-by: Rebecca Dimock <66339736+beckykd@users.noreply.github.com> --- docs/tutorials/ghz-spacetime-codes.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index 5ed09c2fb615..e52d888478e6 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -62,7 +62,7 @@ "- `NoiseModel.from_backend` builds a rough noise model from backend benchmark data. The model scores candidate checks, so an exact, learned noise model is not required. For a learned Pauli-Lindblad model, see `NoiseModel.from_pauli_lindblad_maps`.\n", "- `add_pauli_checks` finds and inserts checks into a circuit. It returns a sequence of `CheckedCircuit` objects with an increasing number of checks, and each object provides a `get_postselection_method` that maps a measured bitstring to a syndrome vector. The `cost` argument selects the function that scores a check (`gamma`, the sampling overhead of the postselected inverse noise channel, or `LER`, the logical error rate). The `method` argument selects the search strategy (`windowed`, `genetic`, or `windowed_genetic`). This tutorial uses `cost=\"gamma\"` and `method=\"windowed\"`, which together give deterministic, reproducible check selection.\n", "\n", - "### Estimating fidelity from stabilizer sampling\n", + "### Estimate fidelity from stabilizer sampling\n", "\n", "To measure how well the error detection works, you can estimate the fidelity of the stabilizer state $|\\psi\\rangle = U|0\\rangle^{\\otimes n}$ that the circuit ideally prepares against the noisy state $\\rho$ that the hardware actually outputs. The projector onto a pure stabilizer state $|\\psi\\rangle$ equals the uniform average over the $2^n$ elements of its stabilizer group $\\mathcal{S}$:\n", "\n", From 5398a928a00af99fdd52e325496961db6eb5b13f Mon Sep 17 00:00:00 2001 From: Henry Zou <87874865+henryzou50@users.noreply.github.com> Date: Wed, 1 Jul 2026 17:11:57 -0400 Subject: [PATCH 05/14] Update docs/tutorials/ghz-spacetime-codes.ipynb Co-authored-by: Rebecca Dimock <66339736+beckykd@users.noreply.github.com> --- docs/tutorials/ghz-spacetime-codes.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index e52d888478e6..c732e3762fa1 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -72,7 +72,7 @@ "\n", "$$F = \\mathrm{Tr}(\\rho|\\psi\\rangle\\langle\\psi|) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\mathrm{Tr}(\\rho G) = \\frac{1}{2^n} \\sum_{G \\in \\mathcal{S}} \\langle G \\rangle_\\rho.$$\n", "\n", - "For larger problems, enumerating all $2^n$ stabilizers is infeasible, so you can estimate the fidelity from a random sample. Drawing $M$ stabilizers $G_1, \\ldots, G_M$ uniformly at random from $\\mathcal{S}$ gives an unbiased estimator:\n", + "For larger problems, enumerating all $2^n$ stabilizers is infeasible, so you can estimate the fidelity from a random sample. Drawing $M$ stabilizers $G_1, \\ldots, G_M$ uniformly at random from $\\mathcal{S}$ gives an unbiased estimation:\n", "\n", "$$\\hat F_M = \\frac{1}{M} \\sum_{i=1}^M \\langle G_i \\rangle_\\rho.$$\n", "\n", From 92c4931bf1c910149429863130538c17678c444a Mon Sep 17 00:00:00 2001 From: Henry Zou <87874865+henryzou50@users.noreply.github.com> Date: Wed, 1 Jul 2026 17:12:30 -0400 Subject: [PATCH 06/14] Update docs/tutorials/ghz-spacetime-codes.ipynb Co-authored-by: Rebecca Dimock <66339736+beckykd@users.noreply.github.com> --- docs/tutorials/ghz-spacetime-codes.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index c732e3762fa1..2da8dc6b3e43 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -76,7 +76,7 @@ "\n", "$$\\hat F_M = \\frac{1}{M} \\sum_{i=1}^M \\langle G_i \\rangle_\\rho.$$\n", "\n", - "Because a Clifford circuit prepares a stabilizer state, you can estimate its fidelity directly from sampled expectation values of its stabilizers. This tutorial first walks through the workflow on a simulator with a small circuit, then runs the same workflow on hardware with a deeper circuit. As circuits include more non-Clifford operations the number of valid checks shrinks quickly, so the method works best for Clifford-dominated circuits." + "Because a Clifford circuit prepares a stabilizer state, you can estimate its fidelity directly from sampled expectation values of its stabilizers. This tutorial first walks through the workflow on a simulator with a small circuit, then runs the same workflow on hardware with a deeper circuit. As circuits include more non-Clifford operations, the number of valid checks shrinks quickly, so the method works best for Clifford-dominated circuits." ] }, { From 496dd24b824fdcfe589fa61480d06f0edf80b6d2 Mon Sep 17 00:00:00 2001 From: Henry Zou <87874865+henryzou50@users.noreply.github.com> Date: Wed, 1 Jul 2026 17:12:37 -0400 Subject: [PATCH 07/14] Update docs/tutorials/ghz-spacetime-codes.ipynb Co-authored-by: Rebecca Dimock <66339736+beckykd@users.noreply.github.com> --- docs/tutorials/ghz-spacetime-codes.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index 2da8dc6b3e43..6e0d7ef9d1ac 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -266,7 +266,7 @@ "id": "step2-header", "metadata": {}, "source": [ - "### Step 2: Optimize problem for quantum hardware execution\n", + "### Step 2: Optimize for quantum hardware execution\n", "\n", "Mapping the circuit to hardware sets the physical qubit layout, the noise model that scores candidate checks, and the checks themselves.\n", "\n", From 8562a912c1424e6504ba45305373b537c987b4ed Mon Sep 17 00:00:00 2001 From: Henry Zou Date: Sun, 5 Jul 2026 15:31:09 -0400 Subject: [PATCH 08/14] Make spacetime-codes tutorial backend-agnostic and refresh results Replace the hardcoded ibm_boston qubit layout with a new find_check_layout helper in the Setup section. The helper searches the backend coupling map with random self-avoiding walks, excludes qubits and two-qubit gates whose reported errors exceed configurable thresholds, and keeps the path that offers the most target/ancilla check pairs (ties broken by lower average gate error). Step 2 now selects a device with service.least_busy() and derives the layout from it, so the tutorial runs on any IBM Quantum QPU; the previous fixed layout mapped onto bad edges on non-Boston devices and produced an unusable noise model (~19% gate noise on ibm_marrakesh). Refresh all committed outputs from a successful run on ibm_kingston: - Simulator: fidelity 0.709 -> 0.953 postselected (postselection rate 0.71) - Hardware: fidelity 0.240 -> 0.848 postselected (postselection rate 0.098) Align the commentary with the measured results: - Correct the gamma discussion (the curves fall toward 1.0 but have not converged at seven checks) - Note that the postselection rate approaches the noisy-state fidelity when the checks detect nearly all erroneous samples - Add a short interpretation of the hardware results and the tradeoffs between postselection rate, depth, and number of checks - Reword Step 2 and the transpilation text to be backend-agnostic Finalize the usage estimate (1 minute on a Heron processor, ibm_kingston or equivalent), matching the wording of recently added tutorials. --- docs/tutorials/ghz-spacetime-codes.ipynb | 135 ++++++++++++++---- .../hardware-retrieve-1.avif | Bin 8965 -> 10699 bytes .../extracted-outputs/step2-checks-1.avif | Bin 29206 -> 30023 bytes .../extracted-outputs/step2-layout-1.avif | Bin 10517 -> 10657 bytes .../extracted-outputs/step2-transpile-0.avif | Bin 11621 -> 12358 bytes .../extracted-outputs/step4-fidelity-1.avif | Bin 8181 -> 8984 bytes .../extracted-outputs/step4-fidelity-2.avif | Bin 7997 -> 7421 bytes .../extracted-outputs/step4-gamma-0.avif | Bin 8842 -> 8453 bytes .../extracted-outputs/step4-psr-0.avif | Bin 6472 -> 6441 bytes 9 files changed, 109 insertions(+), 26 deletions(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index 6e0d7ef9d1ac..e960d88a143d 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -12,7 +12,7 @@ "\n", "# Low-overhead error detection with spacetime codes\n", "\n", - "*Usage estimate: 1 minute on a Heron r3 processor (NOTE: This is an estimate only. Your runtime may vary.)*" + "*Usage estimate: 1 minute on a Heron processor (ibm_kingston or equivalent) (NOTE: This is an estimate only. Your runtime may vary.)*" ] }, { @@ -102,7 +102,7 @@ "source": [ "## Setup\n", "\n", - "Import the required libraries and define the helper functions that are not available as imports. The `random_clifford_circuit` function builds a brickwork random Clifford payload, `append_basis_rotation` rotates a circuit so that a stabilizer is measured in the computational basis, `expectation` computes a stabilizer expectation value from sampled counts, and `cum_mean_sem` tracks the running fidelity estimate." + "Import the required libraries and define the helper functions that are not available as imports. The `random_clifford_circuit` function builds a brickwork random Clifford payload, `find_check_layout` searches a backend coupling map for a low-error qubit path with many available ancillas, `append_basis_rotation` rotates a circuit so that a stabilizer is measured in the computational basis, `expectation` computes a stabilizer expectation value from sampled counts, and `cum_mean_sem` tracks the running fidelity estimate." ] }, { @@ -169,6 +169,79 @@ " return qc\n", "\n", "\n", + "def find_check_layout(\n", + " backend,\n", + " num_qubits: int,\n", + " rng: np.random.Generator,\n", + " num_trials: int = 200,\n", + " max_gate_error: float = 0.03,\n", + " max_readout_error: float = 0.2,\n", + ") -> list[int]:\n", + " \"\"\"Find a low-error path of `num_qubits` qubits with many available ancillas.\n", + "\n", + " Builds random self-avoiding walks on the coupling map, excluding the qubits\n", + " and two-qubit gates whose reported errors exceed the thresholds, and keeps\n", + " the path that offers the most target and ancilla pairs. Ties are broken by\n", + " the lower average two-qubit gate error along the path.\n", + " \"\"\"\n", + " target = backend.target\n", + " gate_2q = next(\n", + " name for name in (\"cz\", \"ecr\", \"cx\") if name in target.operation_names\n", + " )\n", + "\n", + " # Collect per-edge gate errors and per-qubit readout errors\n", + " edge_error = {}\n", + " for qubits, props in target[gate_2q].items():\n", + " edge = tuple(sorted(qubits))\n", + " if props is not None and props.error is not None:\n", + " edge_error[edge] = min(edge_error.get(edge, 1.0), props.error)\n", + " readout_error = {\n", + " qubit: target[\"measure\"][(qubit,)].error\n", + " for (qubit,) in target[\"measure\"]\n", + " }\n", + "\n", + " # Keep only the edges whose gate and readout errors are acceptable\n", + " adjacency = {}\n", + " for (q1, q2), error in edge_error.items():\n", + " if (\n", + " error <= max_gate_error\n", + " and readout_error.get(q1, 1.0) <= max_readout_error\n", + " and readout_error.get(q2, 1.0) <= max_readout_error\n", + " ):\n", + " adjacency.setdefault(q1, set()).add(q2)\n", + " adjacency.setdefault(q2, set()).add(q1)\n", + "\n", + " # Random self-avoiding walks; keep the path with the most check pairs\n", + " starts = sorted(adjacency)\n", + " best_path = None\n", + " best_score = (-1, float(\"inf\"))\n", + " for _ in range(num_trials):\n", + " path = [starts[rng.integers(len(starts))]]\n", + " while len(path) < num_qubits:\n", + " options = sorted(adjacency[path[-1]] - set(path))\n", + " if not options:\n", + " break\n", + " path.append(options[rng.integers(len(options))])\n", + " if len(path) < num_qubits:\n", + " continue\n", + " num_pairs = len(get_check_qubits(backend.coupling_map, path)[0])\n", + " mean_error = float(\n", + " np.mean(\n", + " [edge_error[tuple(sorted(e))] for e in zip(path, path[1:])]\n", + " )\n", + " )\n", + " if num_pairs > best_score[0] or (\n", + " num_pairs == best_score[0] and mean_error < best_score[1]\n", + " ):\n", + " best_path, best_score = path, (num_pairs, mean_error)\n", + "\n", + " if best_path is None:\n", + " raise RuntimeError(\n", + " \"No connected low-error path found. Relax the error thresholds.\"\n", + " )\n", + " return best_path\n", + "\n", + "\n", "def append_basis_rotation(\n", " circuit: QuantumCircuit, pauli: Pauli\n", ") -> QuantumCircuit:\n", @@ -270,14 +343,14 @@ "\n", "Mapping the circuit to hardware sets the physical qubit layout, the noise model that scores candidate checks, and the checks themselves.\n", "\n", - "First, choose a one-dimensional qubit layout on the Heron r3 QPU `ibm_boston`. The layout zig-zags diagonally through the coupling graph to maximize the number of available ancilla qubits. The `get_check_qubits` function returns target and ancilla pairs, where a check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", + "First, select a backend and search its coupling map for a one-dimensional qubit layout with the `find_check_layout` helper defined in the Setup section. The helper builds random self-avoiding walks that avoid the highest-error gates and readouts, and it keeps the path that offers the most target and ancilla pairs. Because the search reads the connectivity and error data from the backend itself, the same code runs on any IBM Quantum QPU. The `get_check_qubits` function then returns the target and ancilla pairs, where a check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", "\n", "In the coupling graph that follows, the green qubits are payload qubits and the orange qubits are the ancillas that implement the checks. Qubits with an adjacent ancilla are used as target qubits for checks." ] }, { "cell_type": "code", - "execution_count": 4, + "execution_count": null, "id": "step2-layout", "metadata": {}, "outputs": [ @@ -285,8 +358,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "Target qubits: [68, 69, 89, 91, 111, 113, 133]\n", - "Ancilla qubits: [67, 70, 88, 92, 110, 114, 132]\n" + "Backend: ibm_kingston\n", + "Target qubits: [56, 61, 63, 81, 83, 103, 105]\n", + "Ancilla qubits: [43, 60, 64, 80, 84, 102, 106]\n" ] }, { @@ -302,10 +376,11 @@ ], "source": [ "service = QiskitRuntimeService()\n", - "backend = service.backend(\"ibm_marrakesh\")\n", + "backend = service.least_busy(operational=True, simulator=False)\n", + "print(f\"Backend: {backend.name}\")\n", "\n", - "# Choose a layout and pair each available target qubit with a neighboring ancilla\n", - "layout = [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133]\n", + "# Search for a low-error path, then pair each target qubit with a neighboring ancilla\n", + "layout = find_check_layout(backend, num_qubits, rng)\n", "target_qubits, ancilla_qubits = get_check_qubits(backend, layout)\n", "num_checks = len(target_qubits)\n", "\n", @@ -337,7 +412,7 @@ "id": "step2-transpile-md", "metadata": {}, "source": [ - "With the backend and layout chosen, transpile the payload into an instruction set architecture (ISA) circuit. Setting the layout and translating the gates into the native gate set of `ibm_boston` is enough here." + "With the backend and layout chosen, transpile the payload into an instruction set architecture (ISA) circuit. Setting the layout and translating the gates into the native gate set of the backend is enough here." ] }, { @@ -383,7 +458,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "NoiseModel(gate_noise=0.19074977552838765, readout_noise=0.1016845703125, idling_noise=None)\n" + "NoiseModel(gate_noise=0.0014398050606555474, readout_noise=0.008717854817708334, idling_noise=None)\n" ] } ], @@ -414,7 +489,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Physical layout of payload and ancillas: [68, 69, 78, 89, 90, 91, 98, 111, 112, 113, 119, 133, 67, 70, 88, 92, 110, 114, 132]\n", + "Physical layout of payload and ancillas: [105, 104, 103, 96, 83, 82, 81, 76, 61, 62, 63, 56, 43, 60, 64, 80, 84, 102, 106]\n", "Checked circuit:\n" ] }, @@ -467,7 +542,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {-XYXZIZXYIIXZ, -XZYYIIXZXYYY, ...}\n" + "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {YIYIIIYZXZYX, IYYZYZZZYXYY, ...}\n" ] } ], @@ -520,7 +595,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:14<00:00, 1.37it/s]" + "100%|██████████| 20/20 [00:15<00:00, 1.31it/s]" ] }, { @@ -528,7 +603,7 @@ "output_type": "stream", "text": [ "Added 7 checks to 20 circuits in 15s.\n", - "On average, two-qubit depth increased from 24 to 33 when adding 7 checks.\n" + "On average, two-qubit depth increased from 24 to 32 when adding 7 checks.\n" ] }, { @@ -598,7 +673,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:19<00:00, 1.00it/s]\n" + "100%|██████████| 20/20 [00:20<00:00, 1.01s/it]\n" ] } ], @@ -711,7 +786,7 @@ "id": "step4-fidelity-md", "metadata": {}, "source": [ - "Now compare the fidelity of the bare noisy state with the postselected state. Postselecting only the samples with no detected error raises the expectation value of every stabilizer, and therefore the estimated fidelity. The postselected values use fewer samples than the raw values, yet the expectation values are more accurate and the sampled variance is lower." + "Now compare the fidelity of the bare noisy state with the postselected state. Postselecting only the samples with no detected error raises the expectation value of every stabilizer, and therefore the estimated fidelity. The postselected values use fewer samples than the raw values, yet the expectation values are more accurate and the sampled variance is lower. Notice also that the mean postselection rate is close to the noisy fidelity. This is what you expect when the checks detect nearly all erroneous samples: the fraction of samples that pass every check approaches the fraction of error-free samples, which is the fidelity of the noisy state." ] }, { @@ -725,9 +800,9 @@ "output_type": "stream", "text": [ "ideal fidelity: 1.0\n", - "noisy fidelity: -0.0020\n", - "postselected fidelity: -0.0790\n", - "mean postselection rate: 0.007\n" + "noisy fidelity: 0.7091\n", + "postselected fidelity: 0.9525\n", + "mean postselection rate: 0.709\n" ] }, { @@ -832,7 +907,7 @@ "id": "step4-gamma-md", "metadata": {}, "source": [ - "The gamma score reports how much of the modeled noise channel remains undetected by the checks. Plotting the gamma score against the number of committed checks shows how the detection capability improves as each check is added. A gamma curve that converges toward `1.0` indicates that additional checks detect little extra error, which matches the convergence seen in the postselection rate." + "The gamma score reports how much of the modeled noise channel remains undetected by the checks. Plotting the gamma score against the number of committed checks shows how the detection capability improves as each check is added. A value of `1.0` means that the checks capture all of the modeled noise. The curves fall toward `1.0` as more checks are committed, which shows that each additional check captures part of the remaining undetected error." ] }, { @@ -914,14 +989,14 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 50/50 [03:34<00:00, 4.29s/it]\n" + "100%|██████████| 50/50 [03:41<00:00, 4.42s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Submitted job d91ge2vqq29s738nkm00 with 100 circuits\n" + "Submitted job d95aaqnu62ks7396tahg with 100 circuits\n" ] } ], @@ -989,9 +1064,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "noisy fidelity: 0.0032\n", - "postselected fidelity: 0.0268\n", - "mean postselection rate: 0.008\n" + "noisy fidelity: 0.2397\n", + "postselected fidelity: 0.8475\n", + "mean postselection rate: 0.098\n" ] }, { @@ -1059,6 +1134,14 @@ "plt.show()" ] }, + { + "cell_type": "markdown", + "id": "hardware-results-md", + "metadata": {}, + "source": [ + "On hardware, the payload is deep enough that most samples contain at least one error, so the noisy fidelity is low and postselection discards most shots. The samples that pass every check recover most of the lost fidelity, and the per-stabilizer expectation values separate cleanly from the noisy values. To raise the postselection rate, reduce the circuit depth or commit fewer checks. To tighten the fidelity estimate, increase the number of shots or sampled stabilizers." + ] + }, { "cell_type": "markdown", "id": "next-steps", diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-retrieve-1.avif b/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-retrieve-1.avif index 196275076fb8de3266bef17036b13e60daae15e3..c553a103b9a1cb404da457b93cd709806cc71cbb 100644 GIT binary patch delta 10137 zcmV;KCuZ1%M$1!>cM2#100IC2001eLv4IT%7brh%WMOm?0165jCNW$q1Ozl15Hj&k zlcfRNf1Ib*d*8nnl`3A|NODI=ABzj%(T*P91kd;-k=k(qh zM?*2?SZ*laOSOEFli7SI0Ew}!024*}?$TCtf33dpizp?iu0(0|5_ZI()D()iRlm&> zQ&~QTKCXWl%xxf1d-ZX^pYgDmX~-zjA{1R@2d-TX=L&MExvhMry#Wgr|V; zsfdJ&Sh@(S7Va3v|5>kWT1JC0dz9?njR{rbGX%i}sb$0|A9n}>WjRx&KR0!vBQO)9 zf0+iN=yOEG8;2}K+c(ov0loMM?GCc%j^872i$Qwh2N6DldESXE$mP91j$3SgXk2{5m&5(j0k7EEmJgXXg@Om6cyHEorZ40vn(EwlY5e$o7Qh* z47AtkT2tup4?3&Z@XE^g2Atktu+{=k@x6ma>BLI zLXC>JE?#U^Q6HN1-g9QL+BcI*4JSvH=rLhm@qz@v3hNW>Di?=!LjsJ-%}3hF%b2p 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Co-Authored-By: Claude Fable 5 --- docs/tutorials/ghz-spacetime-codes.ipynb | 168 ++++++++++-------- .../hardware-retrieve-1.avif | Bin 10699 -> 0 bytes .../extracted-outputs/hardware-submit-3.avif | Bin 0 -> 8070 bytes .../extracted-outputs/step2-checks-1.avif | Bin 30023 -> 31354 bytes .../extracted-outputs/step2-layout-1.avif | Bin 10657 -> 10580 bytes .../extracted-outputs/step2-transpile-0.avif | Bin 12358 -> 11943 bytes .../extracted-outputs/step4-fidelity-1.avif | Bin 8984 -> 8988 bytes .../extracted-outputs/step4-fidelity-2.avif | Bin 7421 -> 7564 bytes .../extracted-outputs/step4-gamma-0.avif | Bin 8453 -> 7941 bytes .../extracted-outputs/step4-psr-0.avif | Bin 6441 -> 6090 bytes 10 files changed, 90 insertions(+), 78 deletions(-) delete mode 100644 public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-retrieve-1.avif create mode 100644 public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-3.avif diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index b3941f7c2d5a..051b919392ff 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -12,7 +12,7 @@ "\n", "# Low-overhead error detection with spacetime codes\n", "\n", - "*Usage estimate: 1 minute on a Heron processor (ibm_kingston or equivalent) (NOTE: This is an estimate only. Your runtime may vary.)*" + "*Usage estimate: 4 minutes on a Heron processor (ibm_kingston or equivalent) (NOTE: This is an estimate only. Your runtime may vary.)*" ] }, { @@ -76,7 +76,7 @@ "\n", "$$\\hat F_M = \\frac{1}{M} \\sum_{i=1}^M \\langle G_i \\rangle_\\rho.$$\n", "\n", - "Because a Clifford circuit prepares a stabilizer state, you can estimate its fidelity directly from sampled expectation values of its stabilizers. This tutorial first walks through the workflow on a simulator with a small circuit, then runs the same workflow on hardware with a deeper circuit. As circuits include more non-Clifford operations, the number of valid checks shrinks quickly, so the method works best for Clifford-dominated circuits." + "Because a Clifford circuit prepares a stabilizer state, you can estimate its fidelity directly from sampled expectation values of its stabilizers. This tutorial first walks through the workflow on a simulator with a small circuit, then runs the same workflow on hardware with a larger, deeper circuit. As circuits include more non-Clifford operations, the number of valid checks shrinks quickly, so the method works best for Clifford-dominated circuits." ] }, { @@ -358,9 +358,9 @@ "name": "stdout", "output_type": "stream", "text": [ - "Backend: ibm_kingston\n", - "Target qubits: [56, 61, 63, 81, 83, 103, 105]\n", - "Ancilla qubits: [43, 60, 64, 80, 84, 102, 106]\n" + "Backend: ibm_boston\n", + "Target qubits: [124, 125, 127, 129, 131, 133, 135]\n", + "Ancilla qubits: [123, 117, 137, 118, 138, 119, 139]\n" ] }, { @@ -458,7 +458,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "NoiseModel(gate_noise=0.0014398050606555474, readout_noise=0.008717854817708334, idling_noise=None)\n" + "NoiseModel(gate_noise=0.0011936862049438239, readout_noise=0.00433349609375, idling_noise=None)\n" ] } ], @@ -489,7 +489,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Physical layout of payload and ancillas: [105, 104, 103, 96, 83, 82, 81, 76, 61, 62, 63, 56, 43, 60, 64, 80, 84, 102, 106]\n", + "Physical layout of payload and ancillas: [135, 134, 133, 132, 131, 130, 129, 128, 127, 126, 125, 124, 123, 117, 137, 118, 138, 119, 139]\n", "Checked circuit:\n" ] }, @@ -542,7 +542,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {YIYIIIYZXZYX, IYYZYZZZYXYY, ...}\n" + "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {ZXIIXZYYXIZZ, XXXYIIZYXIII, ...}\n" ] } ], @@ -595,14 +595,14 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:15<00:00, 1.31it/s]" + "100%|██████████| 20/20 [00:14<00:00, 1.39it/s]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Added 7 checks to 20 circuits in 15s.\n", + "Added 7 checks to 20 circuits in 14s.\n", "On average, two-qubit depth increased from 24 to 32 when adding 7 checks.\n" ] }, @@ -673,7 +673,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:20<00:00, 1.01s/it]\n" + "100%|██████████| 20/20 [00:17<00:00, 1.14it/s]\n" ] } ], @@ -800,9 +800,9 @@ "output_type": "stream", "text": [ "ideal fidelity: 1.0\n", - "noisy fidelity: 0.7091\n", - "postselected fidelity: 0.9525\n", - "mean postselection rate: 0.709\n" + "noisy fidelity: 0.8030\n", + "postselected fidelity: 0.9685\n", + "mean postselection rate: 0.793\n" ] }, { @@ -964,9 +964,9 @@ "source": [ "## Large-scale hardware example\n", "\n", - "The same workflow runs on hardware with a deeper payload and more stabilizers. This section reuses the backend, layout, target and ancilla pairs, noise model, and pass manager from the simulator example, then submits the sampled circuits to the QPU in a single job. Because the circuit is deeper, gate noise has a larger effect, and the fidelity gain from postselection is more pronounced.\n", + "The same workflow runs on hardware with a larger, deeper payload. This section reuses the backend from the simulator example but builds a new 20-qubit layout with its own target and ancilla pairs, noise model, and pass manager, then submits the circuits to the QPU in a single job. At this size, most shots trigger at least one check, so the postselection rate is low, and each circuit needs a large shot budget for enough samples to survive. The example therefore concentrates its budget on a few sampled stabilizers - still an unbiased fidelity estimate, just a coarser one than the simulator example's average over many stabilizers.\n", "\n", - "The parameters that follow set the depth, the number of stabilizers, and the number of shots. Increase `hw_num_stabilizers` for a more precise fidelity estimate at the cost of more circuits per job, and adjust the layout to use more qubits if you want a larger payload." + "The parameters that follow set the number of qubits, the depth, the number of stabilizers, and the number of shots. Scale `hw_num_shots` with the inverse of the postselection rate: at a 3% rate, 40,000 shots leave roughly 1,200 postselected samples per circuit. Increase `hw_num_stabilizers` for a more precise fidelity estimate at the cost of more circuits per job, each of which needs the same shot budget." ] }, { @@ -976,7 +976,7 @@ "source": [ "### Steps 1-4 (compressed into a single code block)\n", "\n", - "The first cell builds the deeper payload, samples its stabilizers, finds the fully checked circuit for each one, and submits one Sampler job that contains both the bare and the checked circuits. Each job carries the tag `TUT_ASPC` so that you can find it later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." + "The following cell builds the larger payload, samples a few stabilizers, finds the fully checked circuit for each one, and submits one Sampler job that contains both the bare and the checked circuits. After the job completes, it postselects the checked counts and compares the noisy and postselected fidelity estimates. At this size, enumerating the full stabilizer group like the simulator example is infeasible, so the cell draws each stabilizer directly by composing a random subset of the group generators. Each job carries the tag `TUT_ASPC` so that you can find it later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." ] }, { @@ -985,51 +985,97 @@ "id": "hardware-submit", "metadata": {}, "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Layout with 11 check pairs: [65, 66, 67, 68, 69, 78, 89, 88, 87, 97, 107, 106, 105, 117, 125, 124, 123, 136, 143, 144]\n" + ] + }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 50/50 [03:41<00:00, 4.42s/it]\n" + "100%|██████████| 10/10 [00:37<00:00, 3.79s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Submitted job d95aaqnu62ks7396tahg with 100 circuits\n" + "Submitted job d9ehr72neu4c739ofk40 with 20 circuits\n", + "noisy fidelity estimate: 0.2830\n", + "postselected fidelity estimate: 0.6459\n", + "mean postselection rate: 0.0640 (~2562 kept shots per circuit)\n" ] + }, + { + "data": { + "text/plain": [ + "\"Output" + ] + }, + "metadata": {}, + "output_type": "display_data" } ], "source": [ - "# -------------------------Step 1: build a deeper payload and sample its stabilizers-------------------------\n", - "hw_depth = 72\n", - "hw_num_stabilizers = 50\n", - "hw_num_shots = 2_000\n", + "# -------------------------Step 1: build a larger payload and sample stabilizers-------------------------\n", + "hw_num_qubits = 20\n", + "hw_depth = 36\n", + "hw_num_stabilizers = 10\n", + "hw_num_shots = 40_000\n", "\n", - "hw_circuit = random_clifford_circuit(num_qubits, hw_depth, rng)\n", + "hw_circuit = random_clifford_circuit(hw_num_qubits, hw_depth, rng)\n", "hw_no_meas = hw_circuit.remove_final_measurements(inplace=False)\n", - "hw_group = PauliList([Pauli(\"I\" * num_qubits)])\n", - "for generator in (\n", + "\n", + "# Enumerating all 2^n stabilizers is infeasible at this size, so draw each\n", + "# stabilizer by composing a random subset of the group generators\n", + "hw_generators = [\n", " Pauli(label) for label in Clifford(hw_no_meas).to_labels(mode=\"S\")\n", - "):\n", - " hw_group = hw_group + hw_group.compose(generator)\n", - "keep = np.where(hw_group.x.any(axis=1) | hw_group.z.any(axis=1))[0]\n", - "chosen = np.random.default_rng(seed).choice(\n", - " keep, size=min(hw_num_stabilizers, len(keep)), replace=False\n", + "]\n", + "sample_rng = np.random.default_rng(seed)\n", + "hw_stabilizers = []\n", + "while len(hw_stabilizers) < hw_num_stabilizers:\n", + " mask = sample_rng.integers(0, 2, hw_num_qubits).astype(bool)\n", + " if not mask.any():\n", + " continue # skip the identity\n", + " stabilizer = Pauli(\"I\" * hw_num_qubits)\n", + " for generator, chosen in zip(hw_generators, mask, strict=True):\n", + " if chosen:\n", + " stabilizer = stabilizer.compose(generator)\n", + " hw_stabilizers.append(stabilizer)\n", + "\n", + "# -------------------------Step 2: find a 20-qubit layout, then add the fully checked circuit per stabilizer-------------------------\n", + "# A single bad coupler or bad-readout qubit on the path drags every\n", + "# stabilizer down, so search harder and with tighter error thresholds\n", + "hw_layout = find_check_layout(\n", + " backend,\n", + " hw_num_qubits,\n", + " rng,\n", + " num_trials=500,\n", + " max_gate_error=0.015,\n", + " max_readout_error=0.05,\n", ")\n", - "hw_stabilizers = [hw_group[int(i)] for i in chosen]\n", + "hw_target_qubits, hw_ancilla_qubits = get_check_qubits(backend, hw_layout)\n", + "hw_noise_model = NoiseModel.from_backend(\n", + " backend, hw_layout, uniform_gate_noise=True\n", + ")\n", + "hw_pm = generate_preset_pass_manager(\n", + " optimization_level=0, backend=backend, initial_layout=hw_layout\n", + ")\n", + "print(f\"Layout with {len(hw_target_qubits)} check pairs: {hw_layout}\")\n", "\n", - "# -------------------------Step 2: transpile and add the fully checked circuit per stabilizer-------------------------\n", "hw_noisy_circuits = []\n", "hw_checked_circuits = []\n", "for i, pauli in enumerate(tqdm(hw_stabilizers)):\n", - " bare = pm.run(append_basis_rotation(hw_circuit, pauli))\n", + " bare = hw_pm.run(append_basis_rotation(hw_circuit, pauli))\n", " hw_noisy_circuits.append(bare)\n", " variants = add_pauli_checks(\n", " bare,\n", - " target_qubits,\n", - " noise_model,\n", - " ancilla_qubits=ancilla_qubits,\n", + " hw_target_qubits,\n", + " hw_noise_model,\n", + " ancilla_qubits=hw_ancilla_qubits,\n", " cost=\"gamma\",\n", " method=\"windowed\",\n", " seed=seed + 1 + i,\n", @@ -1043,43 +1089,8 @@ "\n", "pubs = hw_noisy_circuits + [cc.circuit for cc in hw_checked_circuits]\n", "job = sampler.run(pubs)\n", - "print(f\"Submitted job {job.job_id()} with {len(pubs)} circuits\")" - ] - }, - { - "cell_type": "markdown", - "id": "hardware-retrieve-md", - "metadata": {}, - "source": [ - "After the job completes, retrieve the results and postselect the checked counts. Compute the noisy and postselected fidelities as the average stabilizer expectation value over the sampled stabilizers." - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "id": "hardware-retrieve", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "noisy fidelity: 0.2397\n", - "postselected fidelity: 0.8475\n", - "mean postselection rate: 0.098\n" - ] - }, - { - "data": { - "text/plain": [ - "\"Output" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ + "print(f\"Submitted job {job.job_id()} with {len(pubs)} circuits\")\n", + "\n", "# -------------------------Step 4: postselect and compare fidelity-------------------------\n", "result = job.result()\n", "n_stab = len(hw_stabilizers)\n", @@ -1102,9 +1113,10 @@ "hw_fidelity_postsel = float(np.nanmean([r[1] for r in hw_results]))\n", "hw_psr = float(np.mean([r[2] for r in hw_results]))\n", "print(\n", - " f\"noisy fidelity: {hw_fidelity_noisy:.4f}\\n\"\n", - " f\"postselected fidelity: {hw_fidelity_postsel:.4f}\\n\"\n", - " f\"mean postselection rate: {hw_psr:.3f}\"\n", + " f\"noisy fidelity estimate: {hw_fidelity_noisy:.4f}\\n\"\n", + " f\"postselected fidelity estimate: {hw_fidelity_postsel:.4f}\\n\"\n", + " f\"mean postselection rate: {hw_psr:.4f} \"\n", + " f\"(~{int(round(hw_psr * hw_num_shots))} kept shots per circuit)\"\n", ")\n", "\n", "hw_noisy = np.array([r[0] for r in hw_results])\n", @@ -1139,7 +1151,7 @@ "id": "hardware-results-md", "metadata": {}, "source": [ - "On hardware, the payload is deep enough that most samples contain at least one error, so the noisy fidelity is low and postselection discards most shots. The samples that pass every check recover most of the lost fidelity, and the per-stabilizer expectation values separate cleanly from the noisy values. To raise the postselection rate, reduce the circuit depth or commit fewer checks. To tighten the fidelity estimate, increase the number of shots or sampled stabilizers." + "On hardware at this size, most samples contain at least one detected error, so the postselection rate is small and postselection discards most shots. The samples that pass every check give a much higher expectation value than the bare circuit, and the per-stabilizer values separate cleanly from the noisy baseline. To tighten the fidelity estimate, sample more stabilizers with the same per-circuit shot budget. To raise the postselection rate, reduce the circuit depth or commit fewer checks; to push to larger payloads, scale the shot budget with the inverse of the postselection rate." ] }, { diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-retrieve-1.avif b/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-retrieve-1.avif deleted file mode 100644 index c553a103b9a1cb404da457b93cd709806cc71cbb..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 10699 zcmYLvV|1oV)Ab$Ob|&`3wr$(CZQGdGwv&lDv2EMtJLjD5SzrC=+Iv^k>aPCLeXVsh z006)>b@s3~aJ4W6{52a3Q-*)AjfKHq4rFg*;%xAb{?~q(8(Ta62LS+k3nSEz+$Y+~cY`8UVH+QPumgVVs;-u&PF!n3fq`4{{x{o6fA05Avu7&I6d?4Rke zEgVh$YxD1Pe~YdE-JrisQt;Ur8#p5aprNtI7__NDL7A};8E@sJ{ehc&Xfv^0l9CA8 z;f>ieK%AUCFxm4tOq#v$-vMu}cQSCOIxnuyuZxSUuDfk?K_?u1_hUW=<4)1+;>*8m z^mxmND{>HKl-HiRv6CZ56wM6&T#gMBc9ERGq84Z=)zk%W8xz9+6!K^A=De+f;fcQIs@z!+2}_ 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z59@)>Ea(nKl7{ptLS4pQU6Q(dQNwkx+C>-y&aTERRj4~q?synHDxu&+8^!+GV^MCFTq0p_q54iz({Xc!Ex+k ztFg**&Y-<2l7y{Pqtvp?{s6RG8Q9J`t$o87t+w6vNqIgXSo|t_XlIcQ zHNlg@`aa`-6{;)AoGx{uEQPURjAFM#U6Sbq72Il-NJze Date: Mon, 20 Jul 2026 14:44:52 -0400 Subject: [PATCH 11/14] Use a learned noise model for the spacetime-codes hardware example Applied Caleb's suggestion to learn the noise model on the QPU instead of inferring a uniform depolarizing channel from calibration data: - Add a NoiseLearner job that characterizes the 20-qubit layout's entangling layers, and a learned_noise_model helper that converts the learner's PauliLindbladError output (physical, per-layer qubit indices) into the layout-relative PauliLindbladMaps that NoiseModel.from_pauli_lindblad_maps expects. Readout noise still comes from calibration data, since the learner characterizes gate layers only. - Keep the steps 1-4 framing, folding layout search, noise learning, and check-finding into step 2, and note that the cell now submits two jobs (tags TUT_ASPC_LEARN and TUT_ASPC). - Refresh hardware results: noisy 0.37 vs postselected 0.69 at a 28.5% postselection rate (~11.4k kept shots per circuit), up from 6.4% (~2.6k) with the uniform model. Add a per-stabilizer results table (noisy, postselected, PS rate, kept shots) and move the plot legend outside the axes so it cannot hide data points. - Add a comment noting that swapping back to NoiseModel.from_backend is a one-line change for readers who want to compare, clarify the stabilizer-subsampling sentence (Caleb's wording), and pin the backend to ibm_boston in the simulator section. - Refresh simulator outputs from the rerun and update the cspell dictionary/ignore list for new terms. --- docs/tutorials/ghz-spacetime-codes.ipynb | 196 +++++++++++++++--- .../extracted-outputs/hardware-submit-3.avif | Bin 8070 -> 0 bytes .../extracted-outputs/hardware-submit-5.avif | Bin 0 -> 7792 bytes .../extracted-outputs/step2-checks-1.avif | Bin 31354 -> 31294 bytes .../extracted-outputs/step2-layout-1.avif | Bin 10580 -> 10436 bytes .../extracted-outputs/step2-transpile-0.avif | Bin 11943 -> 12161 bytes .../extracted-outputs/step4-fidelity-1.avif | Bin 8988 -> 8952 bytes .../extracted-outputs/step4-fidelity-2.avif | Bin 7564 -> 7561 bytes .../extracted-outputs/step4-gamma-0.avif | Bin 7941 -> 7754 bytes .../extracted-outputs/step4-psr-0.avif | Bin 6090 -> 6288 bytes scripts/config/cspell/dictionaries/qiskit.txt | 1 + 11 files changed, 165 insertions(+), 32 deletions(-) delete mode 100644 public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-3.avif create mode 100644 public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-5.avif diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index 051b919392ff..b788198cd373 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -10,6 +10,8 @@ "description: Automatically insert spacetime Pauli checks with qiskit-paulice and boost sampled Clifford-circuit fidelity by postselecting on check syndromes.\n", "---\n", "\n", + "{/* cspell:ignore postsel, frameon, virt */}\n", + "\n", "# Low-overhead error detection with spacetime codes\n", "\n", "*Usage estimate: 4 minutes on a Heron processor (ibm_kingston or equivalent) (NOTE: This is an estimate only. Your runtime may vary.)*" @@ -102,7 +104,7 @@ "source": [ "## Setup\n", "\n", - "Import the required libraries and define the helper functions that are not available as imports. The `random_clifford_circuit` function builds a brickwork random Clifford payload, `find_check_layout` searches a backend coupling map for a low-error qubit path with many available ancillas, `append_basis_rotation` rotates a circuit so that a stabilizer is measured in the computational basis, `expectation` computes a stabilizer expectation value from sampled counts, and `cum_mean_sem` tracks the running fidelity estimate." + "Import the required libraries and define the helper functions that are not available as imports. The `random_clifford_circuit` function builds a brickwork random Clifford payload, `find_check_layout` searches a backend coupling map for a low-error qubit path with many available ancillas, `learned_noise_model` turns `NoiseLearner` output into a `qiskit-paulice` noise model, `append_basis_rotation` rotates a circuit so that a stabilizer is measured in the computational basis, `expectation` computes a stabilizer expectation value from sampled counts, and `cum_mean_sem` tracks the running fidelity estimate." ] }, { @@ -123,7 +125,7 @@ "\n", "# Qiskit\n", "from qiskit import QuantumCircuit\n", - "from qiskit.quantum_info import Clifford, Pauli, PauliList\n", + "from qiskit.quantum_info import Clifford, Pauli, PauliLindbladMap, PauliList\n", "from qiskit.result import sampled_expectation_value\n", "from qiskit.transpiler import generate_preset_pass_manager\n", "from qiskit.visualization import plot_coupling_map\n", @@ -134,7 +136,7 @@ "from qiskit_aer.noise import ReadoutError, depolarizing_error\n", "\n", "# Qiskit IBM Runtime\n", - "from qiskit_ibm_runtime import QiskitRuntimeService\n", + "from qiskit_ibm_runtime import NoiseLearner, QiskitRuntimeService\n", "from qiskit_ibm_runtime import SamplerV2 as Sampler\n", "\n", "# Qiskit Paulice\n", @@ -242,6 +244,52 @@ " return best_path\n", "\n", "\n", + "def learned_noise_model(layer_errors, layout: list[int]) -> NoiseModel:\n", + " \"\"\"Build a `NoiseModel` from `NoiseLearner` results.\n", + "\n", + " `NoiseLearner` reports one `PauliLindbladError` per entangling layer, whose\n", + " generators are indexed against that layer's own physical qubits, while\n", + " `NoiseModel.from_pauli_lindblad_maps` expects `PauliLindbladMap`s indexed the\n", + " way `NoiseModel.from_backend` indexes them: by position in `layout`. This\n", + " translates between the two and drops generators that fall outside `layout`.\n", + " \"\"\"\n", + " phys_to_virt = {phys: virt for virt, phys in enumerate(layout)}\n", + " maps = []\n", + " for layer in layer_errors:\n", + " if layer.error is None:\n", + " continue\n", + " terms = []\n", + " for pauli, rate in zip(\n", + " layer.error.generators, layer.error.rates, strict=True\n", + " ):\n", + " label, indices = [], []\n", + " for local, phys in enumerate(layer.qubits):\n", + " x, z = bool(pauli.x[local]), bool(pauli.z[local])\n", + " if not (x or z):\n", + " continue\n", + " if phys not in phys_to_virt:\n", + " break # generator reaches outside the layout, so skip it\n", + " label.append(\"Y\" if x and z else \"X\" if x else \"Z\")\n", + " indices.append(phys_to_virt[phys])\n", + " else:\n", + " if label:\n", + " terms.append(\n", + " (\"\".join(label), tuple(indices), float(rate))\n", + " )\n", + " # Each map needs a 2-qubit generator to define an entangling layer\n", + " if any(len(t[1]) == 2 for t in terms):\n", + " maps.append(\n", + " PauliLindbladMap.from_sparse_list(\n", + " terms, num_qubits=len(layout)\n", + " )\n", + " )\n", + " if not maps:\n", + " raise RuntimeError(\n", + " \"No usable layer errors. Check that the learner ran on this layout.\"\n", + " )\n", + " return NoiseModel.from_pauli_lindblad_maps(maps)\n", + "\n", + "\n", "def append_basis_rotation(\n", " circuit: QuantumCircuit, pauli: Pauli\n", ") -> QuantumCircuit:\n", @@ -350,7 +398,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 4, "id": "step2-layout", "metadata": {}, "outputs": [ @@ -359,8 +407,8 @@ "output_type": "stream", "text": [ "Backend: ibm_boston\n", - "Target qubits: [124, 125, 127, 129, 131, 133, 135]\n", - "Ancilla qubits: [123, 117, 137, 118, 138, 119, 139]\n" + "Target qubits: [32, 33, 51, 53, 71, 73, 75]\n", + "Ancilla qubits: [31, 34, 50, 54, 70, 79, 59]\n" ] }, { @@ -376,7 +424,9 @@ ], "source": [ "service = QiskitRuntimeService()\n", - "backend = service.least_busy(operational=True, simulator=False)\n", + "backend = service.backend(\"ibm_boston\")\n", + "# backend = service.least_busy(operational=True, simulator=False)\n", + "\n", "print(f\"Backend: {backend.name}\")\n", "\n", "# Search for a low-error path, then pair each target qubit with a neighboring ancilla\n", @@ -458,7 +508,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "NoiseModel(gate_noise=0.0011936862049438239, readout_noise=0.00433349609375, idling_noise=None)\n" + "NoiseModel(gate_noise=0.0013927377378497635, readout_noise=0.005177815755208333, idling_noise=None)\n" ] } ], @@ -489,7 +539,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Physical layout of payload and ancillas: [135, 134, 133, 132, 131, 130, 129, 128, 127, 126, 125, 124, 123, 117, 137, 118, 138, 119, 139]\n", + "Physical layout of payload and ancillas: [75, 74, 73, 72, 71, 58, 51, 52, 53, 39, 33, 32, 31, 34, 50, 54, 70, 79, 59]\n", "Checked circuit:\n" ] }, @@ -542,7 +592,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {ZXIIXZYYXIZZ, XXXYIIZYXIII, ...}\n" + "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {XIXXXYYXXZIZ, ZZZXIXXYYIZZ, ...}\n" ] } ], @@ -595,15 +645,15 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:14<00:00, 1.39it/s]" + "100%|██████████| 20/20 [00:14<00:00, 1.35it/s]" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Added 7 checks to 20 circuits in 14s.\n", - "On average, two-qubit depth increased from 24 to 32 when adding 7 checks.\n" + "Added 7 checks to 20 circuits in 15s.\n", + "On average, two-qubit depth increased from 24 to 33 when adding 7 checks.\n" ] }, { @@ -673,7 +723,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:17<00:00, 1.14it/s]\n" + "100%|██████████| 20/20 [00:20<00:00, 1.03s/it]\n" ] } ], @@ -800,9 +850,9 @@ "output_type": "stream", "text": [ "ideal fidelity: 1.0\n", - "noisy fidelity: 0.8030\n", - "postselected fidelity: 0.9685\n", - "mean postselection rate: 0.793\n" + "noisy fidelity: 0.7598\n", + "postselected fidelity: 0.9580\n", + "mean postselection rate: 0.764\n" ] }, { @@ -964,7 +1014,9 @@ "source": [ "## Large-scale hardware example\n", "\n", - "The same workflow runs on hardware with a larger, deeper payload. This section reuses the backend from the simulator example but builds a new 20-qubit layout with its own target and ancilla pairs, noise model, and pass manager, then submits the circuits to the QPU in a single job. At this size, most shots trigger at least one check, so the postselection rate is low, and each circuit needs a large shot budget for enough samples to survive. The example therefore concentrates its budget on a few sampled stabilizers - still an unbiased fidelity estimate, just a coarser one than the simulator example's average over many stabilizers.\n", + "The same workflow runs on hardware with a larger, deeper payload. This section reuses the backend from the simulator example but builds a new 20-qubit layout with its own target and ancilla pairs and pass manager, then submits the circuits to the QPU in a single job. At this size, most shots trigger at least one check, so the postselection rate is low, and each circuit needs a large shot budget for enough samples to survive. The example therefore concentrates its budget on a few sampled stabilizers - still an unbiased fidelity estimate, just a coarser one than the simulator example's average over many stabilizers.\n", + "\n", + "One thing changes relative to the simulator example: instead of inferring a uniform depolarizing channel from calibration data, this section **learns** the noise model with `NoiseLearner` and builds the `qiskit-paulice` model from the result with `NoiseModel.from_pauli_lindblad_maps`. A learned Pauli-Lindblad model captures the spatial structure of the noise on this specific layout, rather than assuming every edge is equally noisy, so the check placement is scored against noise the QPU actually has. This costs an extra job before the sampling job.\n", "\n", "The parameters that follow set the number of qubits, the depth, the number of stabilizers, and the number of shots. Scale `hw_num_shots` with the inverse of the postselection rate: at a 3% rate, 40,000 shots leave roughly 1,200 postselected samples per circuit. Increase `hw_num_stabilizers` for a more precise fidelity estimate at the cost of more circuits per job, each of which needs the same shot budget." ] @@ -976,7 +1028,9 @@ "source": [ "### Steps 1-4 (compressed into a single code block)\n", "\n", - "The following cell builds the larger payload, samples a few stabilizers, finds the fully checked circuit for each one, and submits one Sampler job that contains both the bare and the checked circuits. After the job completes, it postselects the checked counts and compares the noisy and postselected fidelity estimates. At this size, enumerating the full stabilizer group like the simulator example is infeasible, so the cell draws each stabilizer directly by composing a random subset of the group generators. Each job carries the tag `TUT_ASPC` so that you can find it later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." + "The following cell runs the same four steps as the simulator example. It builds the larger payload and samples a few stabilizers (step 1); chooses a layout, learns the noise model on it, and finds the fully checked circuit for each stabilizer (step 2); submits one Sampler job that contains both the bare and the checked circuits (step 3); and postselects the checked counts to compare the noisy and postselected fidelity estimates, per stabilizer and on average (step 4). At this size, enumerating the full stabilizer group like the simulator example is infeasible, so the cell subsamples a random subset of stabilizers to calculate an estimate of the fidelity.\n", + "\n", + "Note that step 2 does more here than in the simulator example: learning the noise model submits its own `NoiseLearner` job before the Sampler job, so the cell runs **two** jobs in total. They carry the tags `TUT_ASPC_LEARN` and `TUT_ASPC` so that you can find them later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." ] }, { @@ -989,30 +1043,59 @@ "name": "stdout", "output_type": "stream", "text": [ - "Layout with 11 check pairs: [65, 66, 67, 68, 69, 78, 89, 88, 87, 97, 107, 106, 105, 117, 125, 124, 123, 136, 143, 144]\n" + "Layout with 11 check pairs: [153, 152, 151, 138, 131, 130, 129, 118, 109, 110, 111, 98, 91, 90, 89, 78, 69, 70, 71, 58]\n", + "Submitted noise-learner job d9f4mncjeosc73fjfmkg\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/henryzou/.venvs/qiskit-doc-all-dev/lib/python3.14/site-packages/qiskit/qpy/interface.py:388: UserWarning: The qiskit version used to generate the provided QPY file, 2.5.0, is newer than the current qiskit version 2.4.1. This may result in an error if the QPY file uses instructions not present in this current qiskit version\n", + " warnings.warn(\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Learned 4 layers; readout noise 0.00470\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 10/10 [00:37<00:00, 3.79s/it]\n" + "100%|██████████| 10/10 [01:01<00:00, 6.18s/it]\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ - "Submitted job d9ehr72neu4c739ofk40 with 20 circuits\n", - "noisy fidelity estimate: 0.2830\n", - "postselected fidelity estimate: 0.6459\n", - "mean postselection rate: 0.0640 (~2562 kept shots per circuit)\n" + "Submitted job d9f4s04jeosc73fjftkg with 20 circuits\n", + "noisy fidelity estimate: 0.3685\n", + "postselected fidelity estimate: 0.6869\n", + "mean postselection rate: 0.2851 (~11404 kept shots per circuit)\n", + "\n", + "per-stabilizer results:\n", + "idx noisy postsel psr kept shots\n", + " 0 0.3769 0.6918 0.3247 12987\n", + " 1 0.3745 0.6760 0.2999 11995\n", + " 2 0.3659 0.6389 0.3549 14196\n", + " 3 0.3821 0.7060 0.2660 10641\n", + " 4 0.3653 0.7475 0.2531 10124\n", + " 5 0.3752 0.7022 0.2698 10791\n", + " 6 0.3508 0.7144 0.2711 10842\n", + " 7 0.3485 0.7087 0.2381 9523\n", + " 8 0.3825 0.6289 0.2928 11711\n", + " 9 0.3630 0.6549 0.2808 11232\n" ] }, { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "metadata": {}, @@ -1046,7 +1129,7 @@ " stabilizer = stabilizer.compose(generator)\n", " hw_stabilizers.append(stabilizer)\n", "\n", - "# -------------------------Step 2: find a 20-qubit layout, then add the fully checked circuit per stabilizer-------------------------\n", + "# -------------------------Step 2: find a 20-qubit layout, learn its noise, and add checks-------------------------\n", "# A single bad coupler or bad-readout qubit on the path drags every\n", "# stabilizer down, so search harder and with tighter error thresholds\n", "hw_layout = find_check_layout(\n", @@ -1058,14 +1141,47 @@ " max_readout_error=0.05,\n", ")\n", "hw_target_qubits, hw_ancilla_qubits = get_check_qubits(backend, hw_layout)\n", - "hw_noise_model = NoiseModel.from_backend(\n", - " backend, hw_layout, uniform_gate_noise=True\n", - ")\n", "hw_pm = generate_preset_pass_manager(\n", " optimization_level=0, backend=backend, initial_layout=hw_layout\n", ")\n", "print(f\"Layout with {len(hw_target_qubits)} check pairs: {hw_layout}\")\n", "\n", + "# ----- learn a Pauli-Lindblad noise model on this layout -----\n", + "# The simulator example scored checks against a uniform depolarizing channel\n", + "# inferred from calibration data. Here, learn the noise instead: NoiseLearner\n", + "# runs its own job on the QPU and returns a Pauli-Lindblad channel per unique\n", + "# entangling layer, so the checks are placed against the noise this layout\n", + "# actually has, including its spatial structure. All the sampled stabilizers\n", + "# share the same entangling layers and differ only in their final basis\n", + "# rotation, so learning on the bare payload covers all of them.\n", + "learner = NoiseLearner(\n", + " mode=backend,\n", + " options={\n", + " \"max_layers_to_learn\": 4,\n", + " \"num_randomizations\": 32,\n", + " \"shots_per_randomization\": 128,\n", + " \"environment\": {\"job_tags\": [\"TUT_ASPC_LEARN\"]},\n", + " },\n", + ")\n", + "learner_job = learner.run([hw_pm.run(hw_circuit)])\n", + "print(f\"Submitted noise-learner job {learner_job.job_id()}\")\n", + "hw_layer_errors = learner_job.result().data\n", + "\n", + "# To see how much the learned model helps, swap the next line for the\n", + "# simulator example's uniform model - a one-line change:\n", + "# hw_noise_model = NoiseModel.from_backend(backend, hw_layout, uniform_gate_noise=True)\n", + "hw_noise_model = learned_noise_model(hw_layer_errors, hw_layout)\n", + "# NoiseLearner characterizes gate noise only, so keep the readout estimate\n", + "# from calibration data rather than leaving it unset\n", + "hw_noise_model.readout_noise = NoiseModel.from_backend(\n", + " backend, hw_layout, uniform_gate_noise=True\n", + ").readout_noise\n", + "print(\n", + " f\"Learned {len(hw_layer_errors)} layers; \"\n", + " f\"readout noise {hw_noise_model.readout_noise:.5f}\"\n", + ")\n", + "\n", + "# ----- add the fully checked circuit per stabilizer -----\n", "hw_noisy_circuits = []\n", "hw_checked_circuits = []\n", "for i, pauli in enumerate(tqdm(hw_stabilizers)):\n", @@ -1119,11 +1235,25 @@ " f\"(~{int(round(hw_psr * hw_num_shots))} kept shots per circuit)\"\n", ")\n", "\n", + "# Per-stabilizer breakdown. The postselection rate varies from stabilizer to\n", + "# stabilizer, so a stabilizer whose postselected value barely moves is usually\n", + "# one whose checks rejected little; the kept-shot count says how much of the\n", + "# gap is statistics rather than signal.\n", + "print(\"\\nper-stabilizer results:\")\n", + "print(\n", + " f\"{'idx':>3} {'noisy':>8} {'postsel':>8} {'psr':>7} {'kept shots':>10}\"\n", + ")\n", + "for i, (noisy, post, psr_i) in enumerate(hw_results):\n", + " print(\n", + " f\"{i:>3} {noisy:>8.4f} {post:>8.4f} {psr_i:>7.4f} \"\n", + " f\"{int(round(psr_i * hw_num_shots)):>10}\"\n", + " )\n", + "\n", "hw_noisy = np.array([r[0] for r in hw_results])\n", "hw_post = np.array([r[1] for r in hw_results])\n", "idx = np.arange(n_stab)\n", "\n", - "fig, ax = plt.subplots(figsize=(8, 4))\n", + "fig, ax = plt.subplots(figsize=(9, 4))\n", "ax.axhline(1.0, color=\"black\", linewidth=1.5, label=\"ideal\")\n", "strip(ax, hw_noisy, \"red\", \"noisy\")\n", "strip(ax, hw_post, \"green\", \"postselected\")\n", @@ -1139,8 +1269,10 @@ "ax.set_xlabel(\"stabilizer index\")\n", "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", "ax.set_ylim((-0.1, 1.1))\n", + "ax.set_xticks(idx)\n", "ax.set_title(\"Per-stabilizer expectation values on hardware\")\n", - "ax.legend(loc=\"lower left\")\n", + "# Outside the axes so it cannot hide a data point\n", + "ax.legend(loc=\"center left\", bbox_to_anchor=(1.02, 0.5), frameon=False)\n", "ax.grid(True, alpha=0.3)\n", "plt.tight_layout()\n", "plt.show()" diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-3.avif b/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-3.avif deleted file mode 100644 index e01132506e7a4d8e67d4843c0fa4514e8cae487b..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 8070 zcmYLubyyVM^YzjQNJzJImxO?HOTz-vu=J8kcL^fh-67p6-JQ}PQj*f_()mL_@ArA< zk2~j_ne)t@`|kw+0Av;}o)C~5*aGkpc3=yxzt|28df{*oJ98J%U;QPhtjuhk{(}Gj z1Z?W^Km5Op2?4v<{UdNMClA=x!SpYa5(B^k{we^t#Q!n?K;H49y#%G&{}cf43*`rc z?f+flUlsp_nEz{gaf94Ax&Au-zxY!5pI8lsI6+>D;b4fV!%M~hxp7GbB4YeC;DMbX z=6?YIfI#qahmaS54FP+A|0$@bs4w~d^^^UTUbgtJ2KTqZ4|4U8{42XU*$di1%q{;N z5`v~+Q;49mr?ZQ>owMLe57-tAa`F@e*+Q)Ty%#wcV)qZcSYNhB2!uxfz#|63BmEsu z0(LV0Z_Sr+FJ{|+5A?Dm`JJ5^$mIW1k%t8PUBw5*0Qd%9whR*2gapbK zOtbeo2+l5^MErRI=8fJI_kg?aP^H&+1~+alxAXISZkx^QLHh!tuu)&om~$k*++w<& zkw^(mSvJ-W)m(3?DZC1tQMT#1eg%}E__|ro?%<7~DN%#=;mHJ=rgb*1^S-nt1eg-T zQ4U(Od^~hdGP_VX);^HHf|lfzblGS0tTBn@2ZO&hwZ9{!eGGQu@&U*+E|Y~7Rr&K5 zifr=a&1R$MGMegb#A65I@w1(nXqd0|fPV2VJqw?eHY+1$j(h+vc7ylP&@{qiOR#8W z@q+hy7eNxU;hLP2tQ=(o`o1t;Yiq1wG=h2Qi89#((JTmcTG$ao)Kp&~Bd&;T5++6q%CPR`d9JQV zXcHYqT;lDU8+a>bzA1)6>*|8q97V&Cq#c3$BtzB_!P!jjJJphch9YI!)Z;U!!6fEC z%_^?=!cu1aVx3cK0%h$YY(*X+@hezYv?9aabKYgA{hOd*Rrf>{;_L8;<@6-PVxg2| zv(>;ekyjNDyDh=fRI;Ve-1GpPd7quh{S8R82I=xIY^mT&iMbJPa7AoC9Wbw6!N;|# z!-d^AKcU8t2t&J&BlOOy=Fq_C@n+;ZO_fE)dT5M|F6p}rOA#he&0fOSY_Sy21b#*Y zBj($tWa2=e_TFiJ!f<@Ho_Kj$*!MIMBv%giIaqTm{d-qMRP_jwFZ|U6gOqvC*S}Y{ z$ezSUB#0jvirYPAyl0eD)oZ9sUZow=>p$Psz3S!-V{Og+Z+%qvMyM$V4G|`LwC;Vn>o_?My0!O1lv{qNSvGs&%UQ@mzW$hT_F1{VJ zKOXG-mfQrbaEC_>+SEB~`!=e-hn<-{9-?F(GGO+7gBk>MvN5vi1vM>MG#v^dpOOY! zQ8lI0CGTY((*7ylz(giCU(4xi4r)1iU_)&;^8=RH;8>HOxk*MriOW!?wY^WUIZhmr ztxi_nKR{flsLBkeakYim+9o|=_Pz@)CAbcZ?DyJl?ytJc&4;d&K8RD=3lOf9tBL!c zeq-R4`ntiz;T4JuIG?P0<0(wJP8GB_0#|Ba2)p}fm<5--g$+kGT4lya+0cm3P)#Bd&ehec^xh&}d=rr92}A$9--KMW`?md=J5k8FO(g0nW1q0|RFI7jv#{qCM4z30Z6A>zKWB~Rez zYq}eSrdA^LM)LNXwJx%2->n_(yke@vxb>B$wYzrSsr`nJ*hADTBmL_(DI&>Mcg*u9xe^{N7JgP`9syjS-q$W;@a9? 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zo7?JtEc{sYP{$rC7)1QMW!#KEW;nLo!XN9D?AnL1Ra)U`22jf?N3r+(Uh`7l3vC{8 zsr{Pv2)9egfdRn;xF*Em+D0nA0$yHPVr12>8rhVf+L^#Ky>g(Q?VptV)_n*Px|RNH z-jQpkv9igTKt4-dbr|U6;CL1S9Wm>r0vFK zPUg<>cq}_c`?go_F>$;Oxq}QBP;`LYYJY3JmQ)mbJooM*v2mAlScxmJfmR!h=AW5%U}6Z@(^Q%m3#?4xg{9A!4zcZ~ zD^?v*vvH$C!C+ql53iH5!jdUx#>-@GihxN6vUEHOKLbygU!;6h@l8l`@ z(O%wxUgRzno0Y6R$cnn+9p0Rfu;PtwHs#(CrufbMG}-yyPPbr7Dy?<_{#ooz6tcYe7eC5vz*WA Pr@rxqTzt Date: Mon, 20 Jul 2026 14:58:43 -0400 Subject: [PATCH 12/14] Applied Abby's suggestions --- docs/tutorials/ghz-spacetime-codes.ipynb | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index b788198cd373..81efde00fe65 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -14,7 +14,7 @@ "\n", "# Low-overhead error detection with spacetime codes\n", "\n", - "*Usage estimate: 4 minutes on a Heron processor (ibm_kingston or equivalent) (NOTE: This is an estimate only. Your runtime may vary.)*" + "*Usage estimate: 4 minutes on a Heron processor (`ibm_kingston` or equivalent) (NOTE: This is an estimate only. Your runtime may vary.)*" ] }, { @@ -56,12 +56,12 @@ "\n", "The key difference between coherent Pauli checks and spacetime checks is the operators they measure. Coherent Pauli checks measure time-localized, high-weight operators. On qubit topologies with limited connectivity, such as heavy hex, those checks need many SWAP gates and often make the circuit too deep to run in practice. Implementing the checks as spacetime codes instead distributes each check across the payload circuit in space and time. This yields a hardware-efficient encoding that stays effective at detecting logical errors while keeping the qubit and depth overhead low.\n", "\n", - "### What the qiskit-paulice package does\n", + "### What the `qiskit-paulice` package does\n", "\n", "The `qiskit-paulice` package automates the construction of these checks so that you do not have to build them by hand. Its primary role is to find and insert valid spacetime Pauli checks at the locations in a circuit that maximize error detection while minimizing qubit overhead. A check is *valid* when its operators leave the logical action of the payload circuit unchanged, *low-weight* when it uses few entangling gates, and *effective* when it detects a large portion of the errors, relative to the noise the check itself introduces. The package scores candidate checks against a noise model and commits the best ones to the circuit. This tutorial uses three API methods:\n", "\n", "- `get_check_qubits` inspects a backend coupling map and returns target and ancilla qubit pairs. A check on `target_qubits[i]` uses `ancilla_qubits[i]`.\n", - "- `NoiseModel.from_backend` builds a rough noise model from backend benchmark data. The model scores candidate checks, so an exact, learned noise model is not required. For a learned Pauli-Lindblad model, see `NoiseModel.from_pauli_lindblad_maps`.\n", + "- `NoiseModel.from_backend` builds a rough noise model from backend benchmark data. The model scores candidate checks, so an exact, learned noise model is not required. For a learned Pauli-Lindblad model, see [`NoiseModel.from_pauli_lindblad_maps`](/docs/api/qiskit-addon-paulice/qiskit-paulice-noise-models#from_pauli_lindblad_maps).\n", "- `add_pauli_checks` finds and inserts checks into a circuit. It returns a sequence of `CheckedCircuit` objects with an increasing number of checks, and each object provides a `get_postselection_method` that maps a measured bitstring to a syndrome vector. The `cost` argument selects the function that scores a check (`gamma`, the sampling overhead of the postselected inverse noise channel, or `LER`, the logical error rate). The `method` argument selects the search strategy (`windowed`, `genetic`, or `windowed_genetic`). This tutorial uses `cost=\"gamma\"` and `method=\"windowed\"`, which together give deterministic, reproducible check selection.\n", "\n", "### Estimate fidelity from stabilizer sampling\n", @@ -462,7 +462,7 @@ "id": "step2-transpile-md", "metadata": {}, "source": [ - "With the backend and layout chosen, transpile the payload into an instruction set architecture (ISA) circuit. Setting the layout and translating the gates into the native gate set of the backend is enough here." + "With the backend and layout chosen, transpile the payload into an instruction set architecture (ISA) circuit. It is only necessary to set the layout and translate the gates into the native gate set of the backend." ] }, { @@ -1014,7 +1014,7 @@ "source": [ "## Large-scale hardware example\n", "\n", - "The same workflow runs on hardware with a larger, deeper payload. This section reuses the backend from the simulator example but builds a new 20-qubit layout with its own target and ancilla pairs and pass manager, then submits the circuits to the QPU in a single job. At this size, most shots trigger at least one check, so the postselection rate is low, and each circuit needs a large shot budget for enough samples to survive. The example therefore concentrates its budget on a few sampled stabilizers - still an unbiased fidelity estimate, just a coarser one than the simulator example's average over many stabilizers.\n", + "The same workflow runs on hardware with a larger, deeper payload. This section reuses the backend from the simulator example but builds a new 20-qubit layout with its own target and ancilla pairs and pass manager, then submits the circuits to the QPU in a single job. At this size, most shots trigger at least one check, so the postselection rate is low, and each circuit needs a large shot budget for enough samples to survive. The example therefore concentrates its budget on a few sampled stabilizers; this is still an unbiased fidelity estimate, but is coarser than the simulator example's average over many stabilizers.\n", "\n", "One thing changes relative to the simulator example: instead of inferring a uniform depolarizing channel from calibration data, this section **learns** the noise model with `NoiseLearner` and builds the `qiskit-paulice` model from the result with `NoiseModel.from_pauli_lindblad_maps`. A learned Pauli-Lindblad model captures the spatial structure of the noise on this specific layout, rather than assuming every edge is equally noisy, so the check placement is scored against noise the QPU actually has. This costs an extra job before the sampling job.\n", "\n", @@ -1030,7 +1030,7 @@ "\n", "The following cell runs the same four steps as the simulator example. It builds the larger payload and samples a few stabilizers (step 1); chooses a layout, learns the noise model on it, and finds the fully checked circuit for each stabilizer (step 2); submits one Sampler job that contains both the bare and the checked circuits (step 3); and postselects the checked counts to compare the noisy and postselected fidelity estimates, per stabilizer and on average (step 4). At this size, enumerating the full stabilizer group like the simulator example is infeasible, so the cell subsamples a random subset of stabilizers to calculate an estimate of the fidelity.\n", "\n", - "Note that step 2 does more here than in the simulator example: learning the noise model submits its own `NoiseLearner` job before the Sampler job, so the cell runs **two** jobs in total. They carry the tags `TUT_ASPC_LEARN` and `TUT_ASPC` so that you can find them later. See [Add job tags](/docs/guides/add-job-tags) for more on tagging jobs." + "Note that step 2 does more here than in the simulator example: learning the noise model submits its own `NoiseLearner` job before the Sampler job, so the cell runs **two** jobs in total. They carry the tags `TUT_ASPC_LEARN` and `TUT_ASPC` so that you can find them later. See [Organize and search by job tags](/docs/guides/add-job-tags) for more on tagging jobs." ] }, { From 6eafb8713c2fffcdc9faef32623e3086ebd89574 Mon Sep 17 00:00:00 2001 From: Henry Zou Date: Mon, 27 Jul 2026 12:12:31 -0400 Subject: [PATCH 13/14] Applied Caleb's review suggestions - Rephrase the learned-noise prose: checks are scored against "noise more similar to that affecting the QPU", and learning the noise requires QPU sampling that belongs in the overall sampling budget. - "On hardware at this size" -> "For circuits of this size", and "much higher expectation value" -> "much better expectation value". - Add the dashed mean lines to the per-stabilizer plot legends as "noisy/postselected fidelity" and relabel the points as "noisy/postselected EVs", in both the simulator and hardware plots. - Restore least_busy backend selection and refresh the simulator outputs and images, which also picks up the new legend labels. - Regenerate the hardware plot image from the saved run's per-stabilizer values (no new QPU job; results unchanged) and drop an environment-specific QPY version warning from the saved output. Keep NoiseLearner and Sampler as-is: Sampler is already SamplerV2 via the aliased import, and NoiseLearnerV3 pairs with the Executor stack rather than the V2 primitives this tutorial uses. Slug rename to spacetime-codes will follow in a separate commit. --- docs/tutorials/ghz-spacetime-codes.ipynb | 49 ++++++++---------- .../extracted-outputs/hardware-submit-5.avif | Bin 7792 -> 8662 bytes .../extracted-outputs/step2-checks-1.avif | Bin 31294 -> 33447 bytes .../extracted-outputs/step2-layout-1.avif | Bin 10436 -> 10567 bytes .../extracted-outputs/step2-transpile-0.avif | Bin 12161 -> 12349 bytes .../extracted-outputs/step4-fidelity-1.avif | Bin 8952 -> 10022 bytes .../extracted-outputs/step4-fidelity-2.avif | Bin 7561 -> 7531 bytes .../extracted-outputs/step4-gamma-0.avif | Bin 7754 -> 8655 bytes .../extracted-outputs/step4-psr-0.avif | Bin 6288 -> 6327 bytes 9 files changed, 21 insertions(+), 28 deletions(-) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/ghz-spacetime-codes.ipynb index 81efde00fe65..8d95d932a65e 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/ghz-spacetime-codes.ipynb @@ -398,7 +398,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": null, "id": "step2-layout", "metadata": {}, "outputs": [ @@ -407,8 +407,8 @@ "output_type": "stream", "text": [ "Backend: ibm_boston\n", - "Target qubits: [32, 33, 51, 53, 71, 73, 75]\n", - "Ancilla qubits: [31, 34, 50, 54, 70, 79, 59]\n" + "Target qubits: [105, 107, 108, 123, 125, 141, 143]\n", + "Ancilla qubits: [104, 97, 109, 122, 126, 140, 144]\n" ] }, { @@ -424,8 +424,7 @@ ], "source": [ "service = QiskitRuntimeService()\n", - "backend = service.backend(\"ibm_boston\")\n", - "# backend = service.least_busy(operational=True, simulator=False)\n", + "backend = service.least_busy(operational=True, simulator=False)\n", "\n", "print(f\"Backend: {backend.name}\")\n", "\n", @@ -508,7 +507,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "NoiseModel(gate_noise=0.0013927377378497635, readout_noise=0.005177815755208333, idling_noise=None)\n" + "NoiseModel(gate_noise=0.001079865281450939, readout_noise=0.006001790364583333, idling_noise=None)\n" ] } ], @@ -539,7 +538,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Physical layout of payload and ancillas: [75, 74, 73, 72, 71, 58, 51, 52, 53, 39, 33, 32, 31, 34, 50, 54, 70, 79, 59]\n", + "Physical layout of payload and ancillas: [108, 107, 106, 105, 117, 125, 124, 123, 136, 143, 142, 141, 104, 97, 109, 122, 126, 140, 144]\n", "Checked circuit:\n" ] }, @@ -592,7 +591,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {XIXXXYYXXZIZ, ZZZXIXXYYIZZ, ...}\n" + "Sampled 20 stabilizers of a 12-qubit circuit with two-qubit depth 24: {ZXIIXZYYXIZZ, XXXYIIZYXIII, ...}\n" ] } ], @@ -645,7 +644,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:14<00:00, 1.35it/s]" + "100%|██████████| 20/20 [00:15<00:00, 1.29it/s]" ] }, { @@ -723,7 +722,7 @@ "name": "stderr", "output_type": "stream", "text": [ - "100%|██████████| 20/20 [00:20<00:00, 1.03s/it]\n" + "100%|██████████| 20/20 [00:17<00:00, 1.13it/s]\n" ] } ], @@ -850,9 +849,9 @@ "output_type": "stream", "text": [ "ideal fidelity: 1.0\n", - "noisy fidelity: 0.7598\n", - "postselected fidelity: 0.9580\n", - "mean postselection rate: 0.764\n" + "noisy fidelity: 0.7899\n", + "postselected fidelity: 0.9679\n", + "mean postselection rate: 0.780\n" ] }, { @@ -908,21 +907,23 @@ " ax.axhspan(\n", " m - s, m + s, color=color, alpha=0.15, label=f\"{label} mean and std\"\n", " )\n", - " ax.axhline(m, color=color, linewidth=1, linestyle=\"--\")\n", + " ax.axhline(\n", + " m, color=color, linewidth=1, linestyle=\"--\", label=f\"{label} fidelity\"\n", + " )\n", "\n", "\n", "fig, ax = plt.subplots(figsize=(8, 4))\n", "ax.axhline(np.nanmean(evs_ideal), color=\"black\", linewidth=1.5, label=\"ideal\")\n", "strip(ax, evs_noisy, \"red\", \"noisy\")\n", "strip(ax, evs_post, \"green\", \"postselected\")\n", - "ax.scatter(idx, evs_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", + "ax.scatter(idx, evs_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy EVs\")\n", "ax.scatter(\n", " idx,\n", " evs_post,\n", " color=\"green\",\n", " s=22,\n", " alpha=0.7,\n", - " label=\"postselected\",\n", + " label=\"postselected EVs\",\n", ")\n", "ax.set_xlabel(\"stabilizer index\")\n", "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", @@ -1016,7 +1017,7 @@ "\n", "The same workflow runs on hardware with a larger, deeper payload. This section reuses the backend from the simulator example but builds a new 20-qubit layout with its own target and ancilla pairs and pass manager, then submits the circuits to the QPU in a single job. At this size, most shots trigger at least one check, so the postselection rate is low, and each circuit needs a large shot budget for enough samples to survive. The example therefore concentrates its budget on a few sampled stabilizers; this is still an unbiased fidelity estimate, but is coarser than the simulator example's average over many stabilizers.\n", "\n", - "One thing changes relative to the simulator example: instead of inferring a uniform depolarizing channel from calibration data, this section **learns** the noise model with `NoiseLearner` and builds the `qiskit-paulice` model from the result with `NoiseModel.from_pauli_lindblad_maps`. A learned Pauli-Lindblad model captures the spatial structure of the noise on this specific layout, rather than assuming every edge is equally noisy, so the check placement is scored against noise the QPU actually has. This costs an extra job before the sampling job.\n", + "One thing changes relative to the simulator example: instead of inferring a uniform depolarizing channel from calibration data, this section **learns** the noise model with `NoiseLearner` and builds the `qiskit-paulice` model from the result with `NoiseModel.from_pauli_lindblad_maps`. A learned Pauli-Lindblad model captures the spatial structure of the noise on this specific layout, rather than assuming every edge is equally noisy, so the check placement is scored against noise more similar to that affecting the QPU. Learning the noise requires sampling the QPU and should be accounted for in any overall QPU sampling budget.\n", "\n", "The parameters that follow set the number of qubits, the depth, the number of stabilizers, and the number of shots. Scale `hw_num_shots` with the inverse of the postselection rate: at a 3% rate, 40,000 shots leave roughly 1,200 postselected samples per circuit. Increase `hw_num_stabilizers` for a more precise fidelity estimate at the cost of more circuits per job, each of which needs the same shot budget." ] @@ -1047,14 +1048,6 @@ "Submitted noise-learner job d9f4mncjeosc73fjfmkg\n" ] }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/henryzou/.venvs/qiskit-doc-all-dev/lib/python3.14/site-packages/qiskit/qpy/interface.py:388: UserWarning: The qiskit version used to generate the provided QPY file, 2.5.0, is newer than the current qiskit version 2.4.1. This may result in an error if the QPY file uses instructions not present in this current qiskit version\n", - " warnings.warn(\n" - ] - }, { "name": "stdout", "output_type": "stream", @@ -1257,14 +1250,14 @@ "ax.axhline(1.0, color=\"black\", linewidth=1.5, label=\"ideal\")\n", "strip(ax, hw_noisy, \"red\", \"noisy\")\n", "strip(ax, hw_post, \"green\", \"postselected\")\n", - "ax.scatter(idx, hw_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy\")\n", + "ax.scatter(idx, hw_noisy, color=\"red\", s=22, alpha=0.7, label=\"noisy EVs\")\n", "ax.scatter(\n", " idx,\n", " hw_post,\n", " color=\"green\",\n", " s=22,\n", " alpha=0.7,\n", - " label=\"postselected\",\n", + " label=\"postselected EVs\",\n", ")\n", "ax.set_xlabel(\"stabilizer index\")\n", "ax.set_ylabel(r\"$\\langle G \\rangle$\")\n", @@ -1283,7 +1276,7 @@ "id": "hardware-results-md", "metadata": {}, "source": [ - "On hardware at this size, most samples contain at least one detected error, so the postselection rate is small and postselection discards most shots. The samples that pass every check give a much higher expectation value than the bare circuit, and the per-stabilizer values separate cleanly from the noisy baseline. To tighten the fidelity estimate, sample more stabilizers with the same per-circuit shot budget. To raise the postselection rate, reduce the circuit depth or commit fewer checks; to push to larger payloads, scale the shot budget with the inverse of the postselection rate." + "For circuits of this size, most samples contain at least one detected error, so the postselection rate is small and postselection discards most shots. The samples that pass every check give a much better expectation value than the bare circuit, and the per-stabilizer values separate cleanly from the noisy baseline. To tighten the fidelity estimate, sample more stabilizers with the same per-circuit shot budget. To raise the postselection rate, reduce the circuit depth or commit fewer checks; to push to larger payloads, scale the shot budget with the inverse of the postselection rate." ] }, { diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-5.avif b/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-5.avif index eaba709b4f0f005ebfa461b8654156555506b227..989e7768e9020459b3adedcab5c16b5e802f40bf 100644 GIT binary patch delta 7575 zcmV;I9cbe4Jk~>ycM2d300IC2001GNv4IT%7a&S)WMOm?0165jCO2#=1Ozl15HkBe zv!wym0)G)K49s*t%2Qpn@jIf@2$#3q1p0NI#h?{+A}*bNh+qr350PwzL){m4ZEswY z`P(lNeaHk9?yCoG=Y0l;NHF!$Zsy$s=LIKg2u&`xI;?dYI!yg^XlZ=Zfhj3&gy%z5 zx)a~k*8k2KSh`f=GTc!`TKw)2KRj^!dr3B@4(yEn zQM(ekz_16J09@{RGr-J!#iMA<{SIv`hH1t0=(SogOzsPA|9G$aae~jSZL~{FhB{4m z`7NZeoK}Nu`%A#K>-mexZ+fD1_CGWc@|;ArB2Ae{{}|@_JiB`1%FQT3g^KnI=v#{W^h#fvqn@wFg!o2i0x-s zU!yVIRQUm7xJ86i6jnk6Ha$LByjE)IU!4U5dVqtAZR0$F-U_)Y_?2+g4cDyl$B607 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z`mrX^3@zb|cioUY`;Q$cTwsJ)^rRH)3+P%iL7Ok6U4gQ)hM*&-I+)~rvscRiI4LVd z07GArk$-@2a*{iU);Us4_w7GLNP{mmd||?B7?S1Bgnf`b$5dgc^0dIf*aLbyfU6TW z8@qi2!*D`O1tfk?4G*r+IC2t*XuZ_egpoS&BxFx@IE{*AnNZ1Hd>9xfR>82oEJMHW zP7C-)m(-qQnIBcyw8>%&@zFJLz(pMI^k%E(J%8>TM-ZG-S0QykdOl-lrfnX~FV6T3Qy*!q1amojN|u97tPw@LdB!pW`J?4)P) zAU|5(T|=wlq3pUdy~Tpv6NKmvK27Kcn~B#{+joSjMkXc8b{niI*t;zRhgPdFMVF}^ z@qdEth;#{|5K0f|RA`K>4acGcNW|)%`X_8x-P0$}w1-}2$5I08xacoNn8AMJKF*4+y-xO++Rnn&lka7iXpy=OmPk-EUN#)R zsNLDMDDW3>UQdybS7|V?CLTiJIVEGv3V%D*@E3c$rWew;7-cn2 zntStq9TK1zO9?O^3NL{Uq#561hj4Zu(?ZkY*`y`e3p^kcfB#9+n&A7aR~umsl7H%f zYy=s?L!Fzeo8N`>RO9XxpQvC(JvT?8CSt^=*ZgV%cz>}VicLUx zT-^L`Qiz&mkq<0r`HkX=l}N7UmlJ}IxCNZH9<=qSPz#GE>jEl0ZQ{!XE5`ahI?gFN zT&2U!u|r7!6@2GmVF#U;Z*r`Y$qdADr}mW_9jeDz8R@Q&U(*XBV;UGPPELs_WbR7> zz86~<2V{U1iDI73#pqq&?^AXw8}Fgk?f^z($LzmE)} z{acNw)UnbW1#DkmWE^bsn(BLt0)8(*jgDcYCR+XnV&~8pTcPa0-EZKzBYOt Date: Mon, 27 Jul 2026 12:24:10 -0400 Subject: [PATCH 14/14] Rename tutorial slug from ghz-spacetime-codes to spacetime-codes Move the notebook and its image directory, and update every reference: image paths in the notebook outputs, the link in the post-selection guide, the tutorials toc and index, the qiskit-bot ownership config, and the notebook-testing config. Note for review: the old URL is already live on production, so this rename needs a platform-side redirect (or acceptance of the broken URL). Same situation as the fractional-gates tutorial rename that was reverted in #5343 - flagging for the docs team to decide. --- docs/guides/post-selection.ipynb | 2 +- docs/tutorials/_toc.json | 2 +- docs/tutorials/index.mdx | 2 +- ...time-codes.ipynb => spacetime-codes.ipynb} | 18 +++++++++--------- .../extracted-outputs/hardware-submit-5.avif | Bin .../extracted-outputs/step1-code-0.avif | Bin .../extracted-outputs/step2-checks-1.avif | Bin .../extracted-outputs/step2-layout-1.avif | Bin .../extracted-outputs/step2-transpile-0.avif | Bin .../extracted-outputs/step4-fidelity-1.avif | Bin .../extracted-outputs/step4-fidelity-2.avif | Bin .../extracted-outputs/step4-gamma-0.avif | Bin .../extracted-outputs/step4-psr-0.avif | Bin qiskit_bot.yaml | 2 +- scripts/config/notebook-testing.toml | 2 +- 15 files changed, 14 insertions(+), 14 deletions(-) rename docs/tutorials/{ghz-spacetime-codes.ipynb => spacetime-codes.ipynb} (97%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/hardware-submit-5.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step1-code-0.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step2-checks-1.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step2-layout-1.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step2-transpile-0.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step4-fidelity-1.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step4-fidelity-2.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step4-gamma-0.avif (100%) rename public/docs/images/tutorials/{ghz-spacetime-codes => spacetime-codes}/extracted-outputs/step4-psr-0.avif (100%) diff --git a/docs/guides/post-selection.ipynb b/docs/guides/post-selection.ipynb index d915b249f719..d5616681aafb 100644 --- a/docs/guides/post-selection.ipynb +++ b/docs/guides/post-selection.ipynb @@ -414,7 +414,7 @@ "\n", " - Understand how to incorporate [noise learning](/docs/guides/noise-learning) into a quantum workload.\n", " - Read through other available [error mitigation and suppression](/docs/guides/error-mitigation-and-suppression-techniques) techniques.\n", - " - Learn how to use [spacetime codes](/docs/tutorials/ghz-spacetime-codes) for a low-overhead approach to error detection\n", + " - Learn how to use [spacetime codes](/docs/tutorials/spacetime-codes) for a low-overhead approach to error detection\n", "" ] } diff --git a/docs/tutorials/_toc.json b/docs/tutorials/_toc.json index c9f7f1b2324e..66dfb5b5eae1 100644 --- a/docs/tutorials/_toc.json +++ b/docs/tutorials/_toc.json @@ -242,7 +242,7 @@ }, { "title": "Low-overhead error detection with spacetime codes", - "url": "/docs/tutorials/ghz-spacetime-codes", + "url": "/docs/tutorials/spacetime-codes", "isNew": true, "isNewDate": "2026-06-29" } diff --git a/docs/tutorials/index.mdx b/docs/tutorials/index.mdx index 3ad267b2f46c..ec7e067c9935 100644 --- a/docs/tutorials/index.mdx +++ b/docs/tutorials/index.mdx @@ -171,7 +171,7 @@ Error detection identifies faulty operations to return noise-free results shot-b * [Repetition codes](/docs/tutorials/repetition-codes) -* [Low-overhead error detection with spacetime codes](/docs/tutorials/ghz-spacetime-codes) +* [Low-overhead error detection with spacetime codes](/docs/tutorials/spacetime-codes) diff --git a/docs/tutorials/ghz-spacetime-codes.ipynb b/docs/tutorials/spacetime-codes.ipynb similarity index 97% rename from docs/tutorials/ghz-spacetime-codes.ipynb rename to docs/tutorials/spacetime-codes.ipynb index 8d95d932a65e..96cd78909c6a 100644 --- a/docs/tutorials/ghz-spacetime-codes.ipynb +++ b/docs/tutorials/spacetime-codes.ipynb @@ -362,7 +362,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "execution_count": 3, @@ -414,7 +414,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "execution_count": 4, @@ -473,7 +473,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "execution_count": 5, @@ -545,7 +545,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "execution_count": 7, @@ -778,7 +778,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "metadata": {}, @@ -857,7 +857,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "metadata": {}, @@ -866,7 +866,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "metadata": {}, @@ -970,7 +970,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "metadata": {}, @@ -1088,7 +1088,7 @@ { "data": { "text/plain": [ - "\"Output" + "\"Output" ] }, "metadata": {}, diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-5.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/hardware-submit-5.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/hardware-submit-5.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/hardware-submit-5.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step1-code-0.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step1-code-0.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step1-code-0.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step1-code-0.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step2-checks-1.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step2-checks-1.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step2-checks-1.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step2-checks-1.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step2-layout-1.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step2-layout-1.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step2-layout-1.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step2-layout-1.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step2-transpile-0.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step2-transpile-0.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step2-transpile-0.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step2-transpile-0.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-fidelity-1.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-fidelity-1.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-fidelity-1.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-fidelity-1.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-fidelity-2.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-fidelity-2.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-fidelity-2.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-fidelity-2.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-gamma-0.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-gamma-0.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-gamma-0.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-gamma-0.avif diff --git a/public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-psr-0.avif b/public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-psr-0.avif similarity index 100% rename from public/docs/images/tutorials/ghz-spacetime-codes/extracted-outputs/step4-psr-0.avif rename to public/docs/images/tutorials/spacetime-codes/extracted-outputs/step4-psr-0.avif diff --git a/qiskit_bot.yaml b/qiskit_bot.yaml index 660c2a1f9ba5..21cd7d27f614 100644 --- a/qiskit_bot.yaml +++ b/qiskit_bot.yaml @@ -672,7 +672,7 @@ notifications: "docs/tutorials/nishimori-phase-transition": - "`@nathanearnestnoble`" - "@kevinsung" - "docs/tutorials/ghz-spacetime-codes": + "docs/tutorials/spacetime-codes": - "`@nathanearnestnoble`" - "`@MeltemTolunay`" - "`@henryzou50`" diff --git a/scripts/config/notebook-testing.toml b/scripts/config/notebook-testing.toml index 001f83bcbed6..c06a2862e744 100644 --- a/scripts/config/notebook-testing.toml +++ b/scripts/config/notebook-testing.toml @@ -173,7 +173,7 @@ notebooks = [ "docs/tutorials/implicit-solvent-calculations.ipynb", "docs/tutorials/simulate-kicked-ising-tem.ipynb", "docs/tutorials/dc-hex-ising.ipynb", - "docs/tutorials/ghz-spacetime-codes.ipynb", + "docs/tutorials/spacetime-codes.ipynb", "docs/tutorials/sml-classification.ipynb", "docs/tutorials/qedma-2d-ising-with-qesem.ipynb", "docs/tutorials/readout-error-mitigation-sampler.ipynb",