diff --git a/docs/guides/cloud-setup-invited.mdx b/docs/guides/cloud-setup-invited.mdx
index f16118ed2018..80111ef6ebd6 100644
--- a/docs/guides/cloud-setup-invited.mdx
+++ b/docs/guides/cloud-setup-invited.mdx
@@ -30,7 +30,7 @@ Your user account is associated with one or more [instances](/docs/guides/instan
* Choose the account that was invited in the email.
* The region controls where your jobs are run and where the job data is kept. You can access either region by using the same API key, but you can only see and access the instances that were created in the region that you're logged in to.
- 
+ 
### 2. Optional: Save your access credentials
diff --git a/docs/guides/cloud-setup.mdx b/docs/guides/cloud-setup.mdx
index a87ff6283fc3..e38bb8558eb0 100644
--- a/docs/guides/cloud-setup.mdx
+++ b/docs/guides/cloud-setup.mdx
@@ -28,7 +28,7 @@ Your user account is associated with one or more [instances](/docs/guides/instan
The region controls where your jobs are run and where the job data is kept. You can access either region by using the same API key, but you can only see and access the instances that were created in the region that you're logged in to.
- 
+ 
### 1. Create an instance
diff --git a/docs/guides/global-data-quantum-optimizer.ipynb b/docs/guides/global-data-quantum-optimizer.ipynb
index e17d1fe786f1..d2f53456fd75 100644
--- a/docs/guides/global-data-quantum-optimizer.ipynb
+++ b/docs/guides/global-data-quantum-optimizer.ipynb
@@ -506,7 +506,7 @@
"id": "560912e7-6722-4f06-87b1-83fd4ac2073f",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
diff --git a/docs/guides/instances.mdx b/docs/guides/instances.mdx
index dd71fa8564bb..6c2dc2a6b999 100644
--- a/docs/guides/instances.mdx
+++ b/docs/guides/instances.mdx
@@ -82,7 +82,7 @@ Follow these steps to create an instance and add it to your account. If you are
* You can access either region by using the same API key, but you can only see and access the instances that were created in the region that you're logged in to.
* If you want to create an Open instance, you must choose the us-east region.
- 
+ 
diff --git a/docs/guides/primitive-input-output.ipynb b/docs/guides/primitive-input-output.ipynb
index efb937fc6201..e07784d71d3a 100644
--- a/docs/guides/primitive-input-output.ipynb
+++ b/docs/guides/primitive-input-output.ipynb
@@ -221,7 +221,7 @@
"\n",
" - *Example 4*: (Standard nd generalization) has a 3x6 parameter value set array and two 3x1 observables array. These combine to create two 3x6 output arrays in a similar manner to the previous example.\n",
"\n",
- ""
+ ""
]
},
{
diff --git a/docs/guides/qiskit-runtime-circuit-timing.ipynb b/docs/guides/qiskit-runtime-circuit-timing.ipynb
index 0eb1191794ae..f554b5b2ea23 100644
--- a/docs/guides/qiskit-runtime-circuit-timing.ipynb
+++ b/docs/guides/qiskit-runtime-circuit-timing.ipynb
@@ -198,7 +198,7 @@
"id": "2788e992",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
diff --git a/learning/courses/basics-of-quantum-information/index.mdx b/learning/courses/basics-of-quantum-information/index.mdx
index 77223c86e5d9..662c42aea1aa 100644
--- a/learning/courses/basics-of-quantum-information/index.mdx
+++ b/learning/courses/basics-of-quantum-information/index.mdx
@@ -5,7 +5,7 @@ description: Learn about quantum information, from states and measurements to qu
# Overview
-
+
Welcome to *Basics of quantum information,* the first course in the *Understanding quantum information and computation* series comprising the following courses:
diff --git a/learning/courses/foundations-of-quantum-error-correction/index.mdx b/learning/courses/foundations-of-quantum-error-correction/index.mdx
index 50dc24326615..81f3175e43d1 100644
--- a/learning/courses/foundations-of-quantum-error-correction/index.mdx
+++ b/learning/courses/foundations-of-quantum-error-correction/index.mdx
@@ -5,7 +5,7 @@ description: Learn how quantum computations can be protected against noise throu
# Overview
-
+
Welcome to *Foundations of Quantum Error Correction,* the fourth course in the *Understanding Quantum Information and Computation* series comprising the following courses:
diff --git a/learning/courses/fundamentals-of-quantum-algorithms/index.mdx b/learning/courses/fundamentals-of-quantum-algorithms/index.mdx
index e88bc5ccf109..acf4d8dc9068 100644
--- a/learning/courses/fundamentals-of-quantum-algorithms/index.mdx
+++ b/learning/courses/fundamentals-of-quantum-algorithms/index.mdx
@@ -5,7 +5,7 @@ description: Learn how quantum algorithms beat classical algorithms for problems
# Overview
-
+
Welcome to *Fundamentals of Quantum Algorithms,* the second course in the *Understanding Quantum Information and Computation* series comprising the following courses:
diff --git a/learning/courses/integrating-quantum-and-high-performance-computing/compute-resources.ipynb b/learning/courses/integrating-quantum-and-high-performance-computing/compute-resources.ipynb
index 27ab30ac232c..cd48764a9ca5 100644
--- a/learning/courses/integrating-quantum-and-high-performance-computing/compute-resources.ipynb
+++ b/learning/courses/integrating-quantum-and-high-performance-computing/compute-resources.ipynb
@@ -49,7 +49,7 @@
"\n",
"Network/connectivity/busses are crucial aspect of any compute infrastructure as they dictate how fast data is transferred between compute components. From LPU to cache of CPU, to RAM, to PCI cards, to network connected devices; all of it is communication and it is crucial to have an accurate mental model of it to design highly optimized algorithms for HPC.\n",
"\n",
- "\n",
+ "\n",
"\n",
"### Scaling classical resources\n",
"\n",
@@ -60,7 +60,7 @@
"* Horizontal scaling: Adding more resources, such as multiple CPUs or GPUs, to work together on a single node or, more commonly, on multiple nodes, enabling distributed computation.\n",
"\n",
"\n",
- "\n",
+ "\n",
"\n",
"\n",
"Some of the scaling concepts from this section will be applicable to the next section on quantum computing resources. Some other aspects of quantum resources will be quantified in new ways.\n",
@@ -154,7 +154,7 @@
"* Vertical scaling would be increasing the number of qubits per chip or improving the fidelity of devices.\n",
"* Horizontal scaling would be connecting chips with couplers or with classical interconnect.\n",
"\n",
- "\n",
+ "\n",
"\n",
"#### Check your understanding\n",
"\n",
@@ -203,7 +203,7 @@
"* Resource provisioning: The process of preparing and making HPC resources available and ready for use by jobs, including hardware and software setup. As we will see later, QPUs are computing resources that can be provisioned similarly to classical HPC resources, with the caveats from the previous section.\n",
"* Job scheduling: The activity of the scheduler software in deciding which jobs run, when, and on which resources, managing priorities and queues to efficiently utilize the HPC system. Although this broad statement applies to quantum resources, there could be less control over timing than with other resources.\n",
"\n",
- "\n",
+ "\n",
"**Example:**\n",
"\n",
"Consider a well-known task as a context for understanding resource management: finding the prime factors of large numbers. Let us further assume that the algorithm being used relies on brute force checking of every potential divisor. While this is often not the most efficient method, it is easy to understand how the workload might be managed.\n",
diff --git a/learning/courses/integrating-quantum-and-high-performance-computing/index.mdx b/learning/courses/integrating-quantum-and-high-performance-computing/index.mdx
index db7b797009f3..99a805f139e3 100644
--- a/learning/courses/integrating-quantum-and-high-performance-computing/index.mdx
+++ b/learning/courses/integrating-quantum-and-high-performance-computing/index.mdx
@@ -5,7 +5,7 @@ description: Discover how quantum and high-performance computing can be combined
# Overview
-
+
This course provides an overview of classical and quantum resources as well hybrid workflows with examples.
diff --git a/learning/courses/integrating-quantum-and-high-performance-computing/next-steps.ipynb b/learning/courses/integrating-quantum-and-high-performance-computing/next-steps.ipynb
index 0a902f1a2a15..e9c37635429e 100644
--- a/learning/courses/integrating-quantum-and-high-performance-computing/next-steps.ipynb
+++ b/learning/courses/integrating-quantum-and-high-performance-computing/next-steps.ipynb
@@ -41,7 +41,7 @@
"\n",
"This strategy of combining integrated technology with a broad user base is how we believe the community will reach quantum advantage in the near future. Quantum advantage is not a single, definitive milestone but a process — a sequence of increasingly robust demonstrations that will be scrutinized, reproduced, and challenged by the community until a scientific consensus is reached. This is the path to demonstrating, by the end of 2026, the first credible and verifiable instances where this new way of computing solves practical problems more efficiently, cost-effectively, or accurately than what is attainable with classical computation alone.\n",
"\n",
- ""
+ ""
]
},
{
@@ -72,7 +72,7 @@
"\n",
"The [IBM Quantum Development Roadmap](https://www.ibm.com/quantum/hardware#roadmap) is a good demonstration of this big picture and these big ideas.\n",
"\n",
- "\n",
+ "\n",
"\n",
"IBM Quantum's hardware roadmap is driven by a focus on increasing qubit scale and connectivity. The Nighthawk series (2025-2028) uses a new square lattice architecture to enhance connectivity, while the Loon processor (2025) introduces \"c-couplers\" to enable non-local qubit connectivity, which is critical for fault-tolerant quantum computing (FTQC). This roadmap culminates in the IBM Quantum Starling (2029) and Blue Jay (2033+) systems, which are designed to deliver large-scale, fault-tolerant computation with millions of gates and thousands of logical qubits.\n",
"\n",
diff --git a/learning/courses/integrating-quantum-and-high-performance-computing/programming-models.ipynb b/learning/courses/integrating-quantum-and-high-performance-computing/programming-models.ipynb
index 1939639b8c61..3c6ba6565ffd 100644
--- a/learning/courses/integrating-quantum-and-high-performance-computing/programming-models.ipynb
+++ b/learning/courses/integrating-quantum-and-high-performance-computing/programming-models.ipynb
@@ -24,11 +24,11 @@
"\n",
"We will start with the programming model for quantum computers. The fundamental programming model which is familiar to nearly all quantum developers is the quantum circuit. We will not get into the details of the quantum circuit model here, as we already have a great [lecture by John Watrous](/learning/courses/basics-of-quantum-information/quantum-circuits/introduction) that explains this in detail. We will only mention that the circuit is built out of a set of lines (called wires) that represent qubits, gates that represent operations on quantum states, and a set of measurements.\n",
"\n",
- "\n",
+ "\n",
"\n",
"Another important programming model concept for quantum computing is what we call computational primitives. These primitives represent some of the most common tasks that users aim to accomplish with a quantum computer. There are several primitives available at the moment, including [Executor](/docs/guides/directed-execution-model#executor-primitive). In this course we will focus primarily on the primitives Sampler and Estimator. Sampler gives you the ability to sample a state prepared by your quantum circuit. It tells you which computational basis states make up the quantum state prepared on your quantum circuit. Estimator allows you to estimate the expectation value of an observable for a system in the state prepared by your quantum circuit. A common context is estimating the energy of a system in a specific state.\n",
"\n",
- "\n",
+ "\n",
"\n",
"The last thing we are going to talk about in this section is transpilation. Transpilation is the process of rewriting a given input circuit to match the physical constraints and Instruction Set Architecture (ISA) of a specific quantum device. Similar to classical compilers, this means translating abstract unitary operations into the native gate set that the target device can execute. It also optimizes the circuit instructions for efficient execution on noisy quantum computers, with the routine gradually changing the circuit's structure by applying several optimization stages.\n",
"\n",
@@ -36,7 +36,7 @@
"\n",
"#### Check your understanding\n",
"\n",
- "How many qubits are in the circuit below? \n",
+ "How many qubits are in the circuit below? \n",
"\n",
"\n",
"\n",
@@ -173,7 +173,7 @@
"\n",
"A concrete example of a task workflow model applied to quantum computing is the [Qiskit patterns framework](/docs/guides/intro-to-patterns). A Qiskit pattern is a general framework designed to break down domain-specific problems into a sequence of stages, especially for quantum tasks. This allows for the seamless composability of new capabilities developed by IBM Quantum® researchers (and others) and enables a future in which quantum computing tasks are performed by powerful heterogeneous (CPU/GPU/QPU) computing infrastructure. The four steps of a Qiskit pattern are mapping, optimization, execution, and post-processing, where all tasks are executed one after another in a pipeline. But with task workflows we are not bound to a linear execution order and can execute tasks in parallel. Each task of a workflow can be an entire parallel job of its own. So, you can mix and match these models to describe arbitrarily complex algorithms, and a workload manager like Slurm will handle these.\n",
"\n",
- "\n",
+ "\n",
"\n",
"The image above illustrates the Qiskit pattern in action. The workflow has a graph structure with four stages. This branch-like structure is orchestrated and executed by the scheduler. The problem is mapped into quantum-executable form (quantum circuit) at the initial stage. In the next stage, this quantum circuit is optimized for the specific quantum hardware. The image shows this as a parallel process, which demonstrates how multiple optimization strategies could be applied at the same time. The optimized quantum circuit is then executed on the actual quantum hardware. This is the third stage of the image where the scheduler works with one purple quantum processing unit. Finally, the results are post-processed by classical resources."
]
@@ -189,7 +189,7 @@
"\n",
"The previous lesson on the SQD workflow mentioned some processes that cannot be parallelized. For example, we need the results of many quantum measurements in order to project our matrix into a subspace of tractable dimension. In turn, we need the diagonalized matrix and the associated state vectors to check self-consistency of the quantum measurements (using, for example, charge conservation). After all that, we need to decide whether the ground state energy has converged sufficiently for our purposes. These steps are necessarily sequential and require testing of convergence and self-consistency conditions before proceeding.\n",
"\n",
- "\n",
+ "\n",
"\n",
"This workflow will be revisited in greater detail and implemented in the next section. The only thing you need to take away from this section is that task workflows are necessary."
]
diff --git a/learning/courses/integrating-quantum-and-high-performance-computing/sqd-skqd.ipynb b/learning/courses/integrating-quantum-and-high-performance-computing/sqd-skqd.ipynb
index a6bd666c9341..3012124855a2 100644
--- a/learning/courses/integrating-quantum-and-high-performance-computing/sqd-skqd.ipynb
+++ b/learning/courses/integrating-quantum-and-high-performance-computing/sqd-skqd.ipynb
@@ -73,7 +73,7 @@
"\n",
"Sample-based Quantum Diagonalization (SQD) is a powerful variational algorithm that implements the subspace method in a quantum way. It avoids costly, complex procedures like Hadamard tests by using a quantum computer to prepare a trial state and sample *bitstrings*, which define the subspace for classical diagonalization.\n",
"\n",
- "\n",
+ "\n",
"\n",
"The SQD algorithm can be broken down into the following steps:\n",
"\n",
@@ -83,7 +83,7 @@
"\n",
"To construct this subspace, we start with an input state $|\\phi_0\\rangle$, such as the Hartree-Fock (HF) state in chemistry. We then apply a parameterized quantum circuit, $U(\\theta)$, which is known as the ansatz.\n",
"\n",
- "\n",
+ "\n",
"\n",
"This diagram illustrates the goal of a good ansatz. The ansatz prepares a quantum state whose support (the set of basis states it is composed of) should ideally have a large overlap with the support of the true ground state. This circuit allows us to quickly project the ansatz onto computational basis states, which will be further used in classical diagonalization. In other words: we don't have to guess an ansatz that is the ground state; we only need it to contain the same basis states. Then classical diagonalization of the projected Hamiltonian will give us the superposition of basis states that best approximates the ground state.\n",
"#### Step 2: Sample the subspace\n",
@@ -203,13 +203,13 @@
"\n",
"To bring it all together, we can represent sample-based algorithms as a combination of different programming models on a set of heterogeneous resources. For example, we can represent our algorithm as a task workflow.\n",
"\n",
- "\n",
+ "\n",
"\n",
"This figure illustrates the fundamental four-stage workflow. At first, we will have a task for quantum circuit preparation that overlaps with our target state, followed by task for transpilation, which only requires classical resources to be executed. Next will be a task that uses primitives to execute our quantum circuit, which requires quantum resources. Finally, we have a post-processing task, which itself might be a parallel diagonalization algorithm running on multiple nodes.\n",
"\n",
"Additionally, we might want to run one of these algorithms many times while we vary our ansatz, or we might want to run them completely in parallel with different populations.\n",
"\n",
- "\n",
+ "\n",
"\n",
"As depicted above, you might run multiple workflows simultaneously while doing the following:\n",
"\n",
diff --git a/learning/courses/quantum-diagonalization-algorithms/krylov.ipynb b/learning/courses/quantum-diagonalization-algorithms/krylov.ipynb
index 0e7f50e45ea6..6d80772a7ecc 100644
--- a/learning/courses/quantum-diagonalization-algorithms/krylov.ipynb
+++ b/learning/courses/quantum-diagonalization-algorithms/krylov.ipynb
@@ -415,7 +415,7 @@
"$$\n",
"Here we use capital $R$ to distinguish the residual from the dimension of our Krylov subspace $r$.\n",
"\n",
- "\n",
+ "\n",
"\n",
"We now want to make a correction step of the form\n",
"$$\n",
@@ -426,7 +426,7 @@
"|e_1\\rangle=|x\\rangle−|x_1\\rangle=|x\\rangle−(|x_0\\rangle+|p_0\\rangle)=|e_0\\rangle−|p_0\\rangle.\n",
"$$\n",
"\n",
- "\n",
+ "\n",
"\n",
"We are interested in how our error behaves when transformed by our matrix. So let us calculate the $A$-norm of the error. That is\n",
"$$\n",
diff --git a/learning/courses/quantum-diagonalization-algorithms/sqd-implementation.ipynb b/learning/courses/quantum-diagonalization-algorithms/sqd-implementation.ipynb
index 06459280e9b0..3e935afbba8a 100644
--- a/learning/courses/quantum-diagonalization-algorithms/sqd-implementation.ipynb
+++ b/learning/courses/quantum-diagonalization-algorithms/sqd-implementation.ipynb
@@ -175,7 +175,7 @@
"id": "0f0bb614",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -191,7 +191,7 @@
"id": "963e8386-39b6-40b7-9740-fffcf1573fe6",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -209,7 +209,7 @@
"id": "2e8ac1d2-8f04-4591-921b-8ba0174e4ad0",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -231,7 +231,7 @@
"id": "d3e24a20-1c86-4aea-8300-13bd2ead4e8f",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -247,7 +247,7 @@
"id": "9afe0036-318c-43b9-9b2c-39a808bda82c",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -265,7 +265,7 @@
"id": "24f54a55-4a09-4508-997a-96306835c7e6",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -288,7 +288,7 @@
"id": "8ec1e433-bc00-42bc-bdbb-288c56c32f9d",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -359,7 +359,7 @@
"\n",
"As IBM hardware has a heavy-hex topology, we will adopt the _zig-zag_ pattern used in [\\[1\\]](#references) and explained above for qubit interactions. In this pattern, orbitals (qubits) with the same spin are connected with a line topology (red and blue circles). Due to the heavy-hex topology, orbitals for different spins have connections between every 4th orbital, that is, the 0th, 4th, 8th, and so on (purple circles).\n",
"\n",
- ""
+ ""
]
},
{
@@ -411,7 +411,7 @@
"id": "7cb99cd9",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -555,7 +555,7 @@
"source": [
"The post-processing part of the SQD workflow can be summarized using the following diagram.\n",
"\n",
- ""
+ ""
]
},
{
@@ -663,7 +663,7 @@
"id": "c1940235",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
diff --git a/learning/courses/quantum-diagonalization-algorithms/sqd-overview.ipynb b/learning/courses/quantum-diagonalization-algorithms/sqd-overview.ipynb
index 25b37f4bcd71..d9a6513849d3 100644
--- a/learning/courses/quantum-diagonalization-algorithms/sqd-overview.ipynb
+++ b/learning/courses/quantum-diagonalization-algorithms/sqd-overview.ipynb
@@ -441,7 +441,7 @@
"id": "2fdcc9e5-ca63-4a97-8a28-330cd81c95dc",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -463,7 +463,7 @@
"id": "1aa0044f-d798-4510-bd67-736a3e574695",
"metadata": {},
"source": [
- ""
+ ""
]
},
{
@@ -503,7 +503,7 @@
"\n",
"Preparing and sampling from any of the above states will generate vectors that have non-zero amplitude in the ground state, and all of them qualify as to have _ground state support_. Note that, sampling $\\vert \\psi_c \\rangle$ includes one extra vector $\\vert 101 \\rangle$ which has $0$ amplitude in the exact ground state. However, earlier we have shown that including such vectors in the subspace is not problematic as the projection and diagonalization operation sets the amplitude of unwanted vectors to $0$, and we can get expected eigenvalue and reconstruct the correct eigenstate.\n",
"\n",
- "\n",
+ "\n",
"\n",
"Thus, preparing and sampling from the exact ground state is not necessary. In fact, doing so can be difficult as the exact ground state is not known _a priori_, and it is often beneficial to **not** prepare and sample from the exact ground state, especially if the wave function (state) is skewed with some basis states having very high probabilities. Consider the following wave function:\n",
"\n",
diff --git a/learning/courses/quantum-diagonalization-algorithms/vqe.ipynb b/learning/courses/quantum-diagonalization-algorithms/vqe.ipynb
index 7d50a02f85db..661226e393d0 100644
--- a/learning/courses/quantum-diagonalization-algorithms/vqe.ipynb
+++ b/learning/courses/quantum-diagonalization-algorithms/vqe.ipynb
@@ -48,7 +48,7 @@
"\n",
"At the core of a VQE problem is an operator that describes a system of interest. We will assume here that the lowest eigenvalue and the corresponding eigenvector of this operator are useful for some scientific or business purpose. Examples might include a chemical Hamiltonian describing a molecule, such that the lowest eigenvalue of the operator corresponds to the ground state energy of the molecule, and the corresponding eigenstate describes the geometry or electron configuration of the molecule. Or the operator could describe a cost of a certain process to be optimized, and the eigenstates could correspond to routes or practices. In some fields, like physics, a \"Hamiltonian\" almost always refers to an operator describing the energy of a physical system. But in quantum computing, it is common to see quantum operators that describe a business or logistical problem also referred to as a \"Hamiltonian\". We will adopt that convention here.\n",
"\n",
- "\n",
+ "\n",
"\n",
"Mapping a physical or optimization problem to qubits is typically a non-trivial task, but those details are not the focus of this course. A general discussion of mapping a problem to a quantum operator can be found in [Quantum computing in practice](/learning/courses/quantum-computing-in-practice). A more detailed look at the mapping of chemistry problems into quantum operators can be found in [Quantum Chemistry with VQE](/learning/courses/quantum-chem-with-vqe).\n",
"\n",
@@ -146,7 +146,7 @@
"\n",
"The word \"ansatz\" is German for \"approach\". The correct plural in German is \"ansätze\", though one often sees \"ansatzes\" or \"ansatze\". In the context of VQE, an ansatz is the quantum circuit you use to create a multi-qubit wave function that most closely approximates the ground state of the system you are studying, and which thus produces the lowest expectation value of your operator. This quantum circuit will contain variational parameters (often collected together in the vector of variables $\\vec{\\theta}$).\n",
"\n",
- "\n",
+ "\n",
"\n",
"An initial set of values $\\vec{\\theta_0}$ of the variational parameters is chosen. We will call the unitary operation of the ansatz on the circuit $U_{\\text{var}}(\\vec{\\theta_0})$. By default, all qubits in IBM® quantum computers are initialized to the $|0\\rangle$ state. When the circuit is run, the state of the qubits will be\n",
"\n",
@@ -211,7 +211,7 @@
"* A 'method' of minimization. This refers to the specific algorithm used to search the parameter space. This is where we would specify, for example, COBYLA or SLSQP.\n",
"* Options. The options available may differ by method. But an example which practically all methods would include is the maximum number of iterations of the optimizer before ending the search: 'maxiter'.\n",
"\n",
- "\n",
+ "\n",
"\n",
"At each iterative step, the expectation value of the Hamiltonian is estimated by making many measurements. This estimated energy is returned by the cost function, and the minimizer updates the information it has about the energy landscape. Exactly what the optimizer does to choose the next step varies from method to method. Some use gradients and select the direction of steepest descent. Others may take noise into account and may require that the cost decrease by a large margin before accepting that the true energy decreases along that direction."
]
@@ -347,7 +347,7 @@
"\n",
"But we cannot measure all those Pauli operators simultaneously. Pauli operators (excluding I) do not commute with each other if they are associated with the same qubit. For example, we can measure `IZIZ` and `ZZXZ` simultaneously, because we can measure I and Z simultaneously for the third qubit, and we can know I and X simultaneously for the first qubit. But we cannot measure `ZZZZ` and `ZZZX` simultaneously, because Z and X do not commute, and both act on the 0th qubit. Experienced readers might recall that two groups of Pauli operators might commute as a set even though each individual qubit's measurements do not commute. Estimator assumes tensor-product Pauli measurements (via basis rotations), corresponding to grouping operators that are qubit-wise commuting. So to simultaneously estimate two strings (A and B) of Pauli operators using Estimator, the Pauli operators of each qubit in A and B must commute. This means we also cannot measure `ZZZZ` and `ZZXX` simultaneously.\n",
"\n",
- "\n",
+ "\n",
"\n",
"So we decompose our matrix $H$ into a sum of Paulis acting on different qubits. Some elements of that sum can be measured all at once; we call this a *group of commuting Paulis*. Depending on how many non-commuting terms there are, we may need many such groups. Call the number of such groups of commuting Pauli strings $N_\\text{GCP}$. If $N_\\text{GCP}$ is small, this could work well. If $H$ has millions of groups, this will not be useful.\n",
"\n",
@@ -355,13 +355,13 @@
"\n",
"You record the expectation value (or the cost function) for the set of parameters $\\vec{\\theta_i}$ used in your state, and then you update the parameters. Over time, you could use the expectation values or cost-function values you’ve estimated to approximate a gradient of your cost function in the subspace of states sampled by your ansatz. Both gradient-based, and gradient-free classical optimizers exist. Both suffer from potential trainability issues, like multiple local minima, and large regions of parameter space with near-zero gradient, called *barren plateaus*.\n",
"\n",
- "\n",
+ "\n",
"\n",
"### 2.3 Factors that determine computational cost\n",
"\n",
"VQE will not solve all your toughest quantum chemistry problems. No. But being better at all calculations is not the point. We have shifted what determines the computational cost.\n",
"\n",
- "\n",
+ "\n",
"\n",
"We’ve shifted from a process whose complexity depends only on matrix dimension to one that depends on required precision and the number of non-commuting Pauli operators that make up the matrix. The last bit has no analog in classical computing.\n",
"\n",
diff --git a/learning/courses/variational-algorithm-design/index.mdx b/learning/courses/variational-algorithm-design/index.mdx
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--- a/learning/courses/variational-algorithm-design/index.mdx
+++ b/learning/courses/variational-algorithm-design/index.mdx
@@ -5,7 +5,7 @@ description: "An overview of variational algorithms: hybrid classical quantum al
# Overview
-
+
This course is intended for individuals who have some experience with quantum computing and want to learn how to write variational algorithms using Qiskit Runtime primitives. It is also suitable for quantum computing professionals who want to expand their knowledge and skills in the field of variational algorithms.
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