Variational quantum eigensolver
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed to approximate the ground-state energy and wavefunction of a quantum system by variationally minimizing the expectation value of its Hamiltonian using a parameterized quantum circuit prepared on a quantum processor, combined with classical optimization techniques.[1] Introduced in 2014 on a photonic quantum processor, VQE leverages the variational principle from quantum mechanics, which guarantees that the computed energy provides an upper bound to the true ground-state energy, making it particularly suitable for noisy intermediate-scale quantum (NISQ) devices with limited coherence times compared to methods like quantum phase estimation.[1][2] Key components of VQE include the ansatz, a parameterized quantum circuit that generates trial wavefunctions (often inspired by classical methods like unitary coupled cluster for chemistry applications); the measurement strategy, which computes the expectation value of the Hamiltonian's Pauli terms on the quantum hardware; and the classical optimizer, such as gradient descent or derivative-free methods like COBYLA, which iteratively adjusts the ansatz parameters to minimize the energy.[2] This iterative process allows VQE to handle complex many-body Hamiltonians that are intractable on classical computers, with demonstrated applications in quantum chemistry for molecules like H₂ and He–H⁺, as well as in condensed matter physics for modeling Ising models and frustrated magnets.[1][2] Since its inception, VQE has evolved with advancements in error mitigation techniques, such as zero-noise extrapolation and readout error correction, to improve accuracy on current hardware, and extensions like adaptive ansatzes that dynamically build circuits to reduce circuit depth and barren plateaus in the optimization landscape.[2] Ongoing research addresses challenges including measurement overhead, noise resilience, and scalability, positioning VQE as a cornerstone for near-term quantum simulations in fields ranging from drug discovery to materials design, though quantum advantage remains contingent on mitigating large prefactors in computational cost.[2]Introduction
Overview
The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm designed to approximate the ground-state energy and corresponding wavefunction of a quantum system by preparing variational trial states using parameterized quantum circuits and minimizing the expectation value of the system's Hamiltonian through classical optimization.[1][3] This approach leverages the variational principle, ensuring that the resulting energy serves as an upper bound to the true ground-state value, while the Hamiltonian—representing the total energy of the system—is encoded into measurable observables on the quantum hardware.[1][3] VQE has emerged as a key method in the Noisy Intermediate-Scale Quantum (NISQ) era, where quantum devices with 50–100 qubits and imperfect gates limit the feasibility of fully fault-tolerant algorithms.[3] It addresses quantum many-body problems, such as simulating molecular electronic structures for quantum chemistry applications, which scale exponentially and become intractable on classical computers for systems beyond a few atoms.[1] By requiring only short-depth circuits and tolerating noise through iterative refinement, VQE enables practical computations on current quantum hardware without the need for error correction.[3] At a high level, the algorithm proceeds by initializing parameters for a quantum ansatz circuit on the quantum processor to generate a trial state, measuring the energy expectation value via repeated executions, and feeding these results to a classical routine that adjusts the parameters to lower the energy until convergence.[1][3] This iterative hybrid loop exploits the strengths of both quantum state preparation and classical optimization, making VQE suitable for exploring ground states in condensed matter physics and beyond.[3]History
The variational quantum eigensolver (VQE) originated in 2014 with the work of Peruzzo et al., who introduced it as a hybrid quantum-classical algorithm to approximate ground-state energies of molecular Hamiltonians using limited quantum resources.[1] Their approach leveraged the variational principle within a photonic quantum processor, enabling practical simulations despite hardware noise.[1] This marked a pivotal shift toward near-term quantum algorithms suitable for noisy intermediate-scale quantum (NISQ) devices. A key early milestone was the first experimental demonstration of VQE on the He–H⁺ molecule in the same 2014 study, where Peruzzo et al. achieved chemical accuracy for bond dissociation energies using a four-qubit setup.[1] In 2016, VQE was extended to the H₂ molecule on superconducting qubit hardware.[4] Subsequent extensions in 2017 by Kandala et al. advanced the method by introducing hardware-efficient ansatze optimized for superconducting quantum processors, alongside readout error mitigation techniques to enhance reliability on multi-qubit systems.[5] Further developments included explorations of unitary coupled-cluster ansatze, which provided chemically inspired trial wavefunctions for improved expressivity, as detailed in the comprehensive review by McArdle et al. From 2018 onward, VQE gained widespread accessibility through integration into open-source frameworks such as IBM's Qiskit (via its Aqua chemistry module) and Google's Cirq paired with OpenFermion, enabling standardized implementations and simulations across diverse hardware backends.[6] The focus on NISQ-era applications intensified after Google's 2019 Sycamore experiment demonstrated quantum advantage, prompting refinements in VQE toward hardware-efficient ansatze that minimize circuit depth and error accumulation on available processors.Theoretical Foundations
Variational Principle
The variational theorem, a cornerstone of quantum mechanics, states that for a Hermitian Hamiltonian operator H with ground-state eigenvector |\psi_0\rangle and corresponding eigenvalue E_0, the expectation value \langle \psi | H | \psi \rangle for any normalized trial state |\psi\rangle satisfies \langle \psi | H | \psi \rangle \geq E_0, with equality holding only if |\psi\rangle = |\psi_0\rangle.[7] This principle, often expressed through the Rayleigh quotient R(\psi) = \frac{\langle \psi | H | \psi \rangle}{\langle \psi | \psi \rangle}, ensures that the ground-state energy minimizes the energy functional over the Hilbert space.[8] The proof relies on the Rayleigh-Ritz method, which approximates the ground state by minimizing the Rayleigh quotient over a finite-dimensional trial manifold spanned by basis functions. Consider a trial state |\phi(\mathbf{a})\rangle = \sum_i a_i |\phi_i\rangle in a subspace of dimension N; the estimated energy is E_{\text{est}}(\mathbf{a}) = \frac{\sum_{i,j} a_i^* a_j \langle \phi_i | H | \phi_j \rangle}{\sum_{i,j} a_i^* a_j \langle \phi_i | \phi_j \rangle}, minimized by solving the generalized eigenvalue problem \det(H - \lambda S) = 0, where H_{ij} = \langle \phi_i | H | \phi_j \rangle and S_{ij} = \langle \phi_i | \phi_j \rangle.[8] The lowest eigenvalue of this matrix provides an upper bound to E_0, as the subspace projection restricts the minimization, and expanding the basis monotonically decreases the estimates toward the exact value from above.[9] In the context of quantum computing, this principle extends to parameterized trial states prepared on quantum hardware, such as variational quantum circuits |\psi(\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta}) |\psi_{\text{init}}\rangle, where U(\boldsymbol{\theta}) is a parameterized unitary operator and \boldsymbol{\theta} denotes tunable parameters.[7] The Rayleigh quotient is then approximated by measuring \langle \psi(\boldsymbol{\theta}) | H | \psi(\boldsymbol{\theta})\rangle on the device, enabling hybrid quantum-classical optimization to approach the ground state within hardware constraints.[8] A key implication for the variational quantum eigensolver (VQE) is its provision of rigorous upper bounds on the ground-state energy, as the variational theorem guarantees that any computed expectation value exceeds or equals E_0, offering a quantifiable measure of approximation quality without requiring full coherence over the system evolution.[7] This property distinguishes VQE from phase estimation methods and facilitates reliable error assessment in noisy intermediate-scale quantum devices.[9]Quantum Hamiltonians and Encoding
In the context of the variational quantum eigensolver (VQE), quantum Hamiltonians relevant to applications like quantum chemistry are typically formulated in second-quantized form to describe fermionic systems, such as electrons in molecules. This representation leverages creation (\hat{a}^\dagger_p) and annihilation (\hat{a}_q) operators that obey anticommutation relations, efficiently capturing the many-body nature of the problem while reducing the Hilbert space dimensionality compared to first quantization. The general form of the electronic Hamiltonian is \hat{H} = \sum_{p,q} h_{pq} \hat{a}^\dagger_p \hat{a}_q + \frac{1}{2} \sum_{p,q,r,s} h_{pqrs} \hat{a}^\dagger_p \hat{a}^\dagger_q \hat{a}_s \hat{a}_r, where the sums run over spin-orbital indices, h_{pq} are one-electron integrals (including kinetic energy and nuclear attraction), and h_{pqrs} are two-electron repulsion integrals, ensuring antisymmetry under particle exchange.[10] This structure arises from the Born-Oppenheimer approximation and Hartree-Fock basis sets, making it central to simulating molecular ground states.[11] To execute VQE on gate-based quantum computers, which operate on qubits, the fermionic Hamiltonian must be mapped to an equivalent operator in the qubit space. This fermion-to-qubit transformation preserves the algebra of the fermionic operators while expressing them as products of Pauli matrices (\hat{I}, \hat{X}, \hat{Y}, \hat{Z}). Two seminal mappings are the Jordan-Wigner and Bravyi-Kitaev transformations, both requiring one qubit per spin-orbital for an N-orbital system, thus incurring a linear qubit overhead in the basis set size. The Jordan-Wigner transformation, originally proposed for spin chains but adapted for fermions in quantum chemistry simulations, encodes each fermionic mode as a qubit and represents creation operators as \hat{a}^\dagger_j = \frac{1}{2} (\hat{X}_j - i \hat{Y}_j) \prod_{k=0}^{j-1} \hat{Z}_k, with a similar form for annihilation operators; this introduces long-range string operators that enforce the Pauli exclusion principle but result in Pauli terms with weights up to O(N), leading to non-local interactions.[11] In contrast, the Bravyi-Kitaev transformation uses a binary tree (or Fenwick tree) structure to balance occupation number and parity encoding, yielding more local operators with typical Pauli weights of O(\log N); for instance, the creation operator takes the form \hat{a}^\dagger_j = \frac{1}{2} \hat{Z}_{P(j)} \otimes (\hat{X}_j - i \hat{Y}_j) \otimes \prod_{u \in U(j)} \hat{X}_u, where P(j) and U(j) denote parity and update indices in the tree, reducing the range of correlations compared to Jordan-Wigner while maintaining exact equivalence.[12] Both transformations ensure the mapped Hamiltonian commutes with the total particle number and spin symmetries when appropriately tapered.[10] Following the mapping, the second-quantized Hamiltonian decomposes into a sum of Pauli strings, the native basis for qubit operators: \hat{H} = \sum_k c_k \hat{P}_k, where each \hat{P}_k is a tensor product of single-qubit Pauli operators across the N qubits, and c_k are real coefficients derived from the integrals h_{pq} and h_{pqrs}. This Pauli encoding facilitates the VQE workflow by allowing the expectation value \langle \hat{H} \rangle to be computed variationally, though the number of terms can grow as O(N^4) for the two-electron part before sparsity exploitation.[10] Jordan-Wigner tends to produce denser expansions with more high-weight terms, exacerbating circuit depth and error accumulation, whereas Bravyi-Kitaev yields sparser, lower-weight decompositions that better suit near-term hardware with limited connectivity.[10] A key challenge in these encodings is the qubit overhead for realistic molecules: for example, simulating heavy elements like iron in biomolecules requires basis sets with hundreds of spin-orbitals, demanding correspondingly many qubits and exceeding current noisy intermediate-scale quantum devices limited to tens of qubits.[11] Additionally, the choice between sparse (e.g., Bravyi-Kitaev, with fewer long-range terms) and denser (e.g., Jordan-Wigner) encodings trades off implementation simplicity against gate efficiency and noise resilience, with sparse variants reducing the overall resource demands but complicating circuit compilation on specific architectures.[10] These mappings thus form the foundational step in preparing chemically relevant problems for VQE, directly influencing algorithmic scalability.[12]Algorithm Components
Ansatz Design
In the variational quantum eigensolver (VQE) algorithm, the ansatz serves as a parameterized quantum circuit that generates a trial wavefunction to approximate the ground state of a target Hamiltonian. This trial state is expressed as |\psi(\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta}) |0\rangle, where U(\boldsymbol{\theta}) is a unitary operator depending on variational parameters \boldsymbol{\theta}, and |0\rangle is typically an initial reference state such as the all-zero computational basis state.[1] The ansatz's role is to provide a flexible, ansatz-dependent manifold of states over which the variational principle minimizes the energy expectation value, enabling the identification of low-energy eigenstates on noisy intermediate-scale quantum devices.[13] Ansatzes are broadly classified into hardware-efficient and problem-inspired types, each tailored to balance computational feasibility and representational power. Hardware-efficient ansatzes prioritize compatibility with current quantum hardware by employing shallow circuits composed of alternating layers of single-qubit rotation gates (e.g., R_x(\theta_i) and R_z(\theta_i)) and native entangling two-qubit gates (e.g., CNOT or CZ), often arranged in a brickwork pattern to minimize gate depth and mitigate noise.[14] This design has demonstrated effectiveness in simulating small molecular Hamiltonians, such as \ce{H2} and \ce{LiH}, achieving chemical accuracy with reduced circuit overhead on superconducting qubit platforms.[14] In contrast, problem-inspired ansatzes incorporate domain-specific structure, such as the unitary coupled-cluster singles and doubles (UCCSD) form, which emulates classical coupled-cluster theory by exponentiating anti-Hermitian operators corresponding to fermionic single and double excitations mapped to Pauli strings via Jordan-Wigner or Bravyi-Kitaev transformations.[15] The UCCSD ansatz, U(\boldsymbol{\theta}) = e^{T(\boldsymbol{\theta}) - T^\dagger(\boldsymbol{\theta})}, where T includes excitation amplitudes, excels in quantum chemistry applications by capturing electron correlation effects with high fidelity.[15] Initial trial functions often begin with a physically motivated reference state to accelerate convergence and improve accuracy. In quantum chemistry contexts, the Hartree-Fock (HF) state—representing a mean-field approximation of the molecular orbital configuration—is commonly used as the starting point, prepared by applying a sequence of gates to encode the occupied orbitals into the qubit register.[13] Adaptive ansatzes, such as the adaptive derivative-assembled pseudo-Trotter (ADAPT) VQE, extend this by incrementally constructing the circuit: operators are selected from a predefined pool (e.g., fermionic excitations) based on their gradient magnitude with respect to the energy, building a compact ansatz layer-by-layer to enhance qubit efficiency and reduce depth for molecules like \ce{LiH}.[16] This approach has shown reductions to fewer than 50% of the parameters of fixed UCCSD forms while maintaining chemical accuracy.[16] Key considerations in ansatz design revolve around the trade-off between expressivity—the ansatz's capacity to span a diverse set of quantum states approaching the true ground state—and trainability, which ensures reliable optimization without encountering barren plateaus where gradients vanish exponentially.[17] Highly expressive ansatzes, such as deep hardware-efficient circuits, risk trainability issues due to concentration of the energy landscape around its mean, leading to exponentially small gradients as system size increases; this phenomenon, analyzed in two-layer circuits, underscores the need for shallower, structured designs.[18] To mitigate barren plateaus, practitioners favor ansatzes with limited depth (e.g., 1-2 layers) or symmetry-preserving elements, ensuring the variational landscape remains navigable for problems up to 20 qubits.[18]Measurement Protocol
In the variational quantum eigensolver (VQE), the measurement protocol involves estimating the expectation value of the encoded Hamiltonian H = \sum_k c_k P_k, where c_k are coefficients and P_k are Pauli strings, to evaluate the energy of a trial state |\psi\rangle. This is achieved by computing \langle H \rangle = \sum_k c_k \langle P_k \rangle, with each \langle P_k \rangle obtained through projective measurements on the quantum hardware. For a given Pauli string P_k, the measurement basis is rotated via single-qubit gates to align with the eigenbasis of P_k, after which the circuit is executed multiple times (shots) to sample measurement outcomes; the expectation value is then estimated from the frequency of +1 and -1 eigenvalues.[1] To mitigate the high circuit depth and measurement overhead associated with measuring each P_k separately, Pauli terms are grouped into sets of mutually commuting operators, allowing simultaneous estimation in a single measurement basis per group. This reduces the number of required quantum circuits from the total number of Pauli terms (often scaling exponentially with system size) to the number of such commuting groups, which can be found using graph coloring algorithms on the commutation graph of the terms. A prominent example is tapered measurements, which exploit Abelian symmetries (e.g., particle number or spin conservation) in the Hamiltonian to eliminate redundant qubits and Pauli terms while preserving the spectrum, further lowering the measurement cost. For instance, in fermionic systems encoded via Jordan-Wigner or Bravyi-Kitaev transformations, up to $2^s terms can be tapered off, where s is the number of independent symmetries.[19] Measurement errors arise primarily from shot noise, which introduces statistical variance inversely proportional to the number of shots per term, and readout errors, where misclassification of qubit states biases the estimates. These can be mitigated using zero-noise extrapolation (ZNE), which amplifies noise artificially (e.g., by inserting idle gates or twirling) and extrapolates the observable to the zero-noise limit via polynomial fitting, improving accuracy without additional hardware assumptions. Symmetry verification techniques complement this by post-selecting on measurement outcomes that respect the system's symmetries, discarding erroneous data and reducing bias from decoherence. The measurement overhead remains a bottleneck, as the number of circuits scales linearly with the number of Pauli groups (typically O(N^4) for N-orbital quantum chemistry Hamiltonians before grouping), requiring thousands to millions of shots for precision. Optimizations like quantum stochastic drift (qDRIFT) address this by probabilistically sampling Pauli terms with probabilities proportional to |c_k|, effectively estimating \langle H \rangle with fewer circuits at the cost of increased shot variance, which scales favorably for sparse or weakly correlated terms. This sampling approach can reduce the effective measurement cost by factors of 10–100 in practice for molecular simulations.Classical Optimization
The classical optimization component of the variational quantum eigensolver (VQE) aims to minimize the variational energy expectation value E(\boldsymbol{\theta}) = \langle \psi(\boldsymbol{\theta}) | \hat{H} | \psi(\boldsymbol{\theta}) \rangle, where \boldsymbol{\theta} denotes the parameters of the quantum ansatz |\psi(\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta}) |0\rangle and \hat{H} is the target Hamiltonian. This minimization leverages the variational principle to approximate the ground-state energy, with the classical routine iteratively updating \boldsymbol{\theta} based on energy evaluations obtained from quantum measurements.[10] The process integrates both gradient-based and derivative-free optimization algorithms, selected based on factors such as noise tolerance, computational cost, and the dimensionality of \boldsymbol{\theta}. For instance, gradient-based methods like Adam, which employs adaptive moment estimates with hyperparameters such as learning rates and momentum coefficients, or L-BFGS, a quasi-Newton approach approximating the Hessian for efficient updates in low-dimensional spaces, require analytical gradients of E(\boldsymbol{\theta}).[10] In contrast, derivative-free methods such as COBYLA, a constrained optimization by linear approximations algorithm that uses simplex-based searches, or Nelder-Mead, which performs direct landscape sampling without derivatives, are particularly robust in noisy quantum environments where gradient estimation may be unreliable.[10][1] A cornerstone for gradient-based optimization in VQE is the parameter-shift rule, which enables exact computation of partial derivatives \partial E / \partial \theta_k through additional quantum circuit evaluations. For Pauli rotation gates (e.g., R_x(\theta), R_y(\theta), R_z(\theta)) with standard generator coefficients, the rule computes the gradient as \frac{\partial \langle \hat{H} \rangle}{\partial \theta_k} = \frac{1}{2} \left[ \langle \hat{H} \rangle (\theta_k + s/2) - \langle \hat{H} \rangle (\theta_k - s/2) \right], where s = \pi is the shift parameter, requiring two shifted evaluations per parameter.[20] This technique avoids numerical differentiation errors and scales linearly with the number of parameters, though it doubles the quantum oracle calls compared to energy evaluations alone; extensions to multi-parameter gates use stochastic sampling for efficiency.[21] The hybrid quantum-classical loop orchestrates this by repeatedly invoking the quantum device as an oracle to compute E(\boldsymbol{\theta}) or shifted expectations, followed by classical updates to \boldsymbol{\theta} until convergence. This iterative feedback, often implemented in frameworks like Qiskit or PennyLane, ensures the ansatz evolves toward lower energies while mitigating quantum hardware limitations.[10] Convergence is typically monitored through criteria such as an energy threshold (e.g., achieving chemical accuracy of 1.6 mHartree relative to the exact ground state) or a small gradient norm (e.g., \|\nabla E(\boldsymbol{\theta})\| < 10^{-6}), halting the optimization when either is satisfied to balance accuracy and resource use.[10] Local minima, a common challenge due to the non-convex energy landscape, are addressed via warm-starting techniques that initialize \boldsymbol{\theta} near promising regions using approximations from classical solvers, prior VQE runs on similar systems, or reduced ansatzes, thereby accelerating convergence and improving solution quality.[22] For example, warm-starting with solutions from simpler molecular geometries has been shown to reduce iteration counts by providing better initial guesses, enhancing overall efficiency in practical VQE applications.Mathematical Formulation
Core Equations
The variational quantum eigensolver (VQE) relies on minimizing a parameterized cost function to approximate the ground-state energy E_0 of a quantum Hamiltonian H. The core cost function is the expectation value of the Hamiltonian with respect to a parameterized trial wavefunction |\psi(\theta)\rangle, given by E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle, assuming the trial state is normalized such that \langle \psi(\theta) | \psi(\theta) \rangle = 1.[23][24] In the more general unnormalized form introduced in the original formulation, it is the Rayleigh quotient E(\theta) = \frac{\langle \psi(\theta) | H | \psi(\theta) \rangle}{\langle \psi(\theta) | \psi(\theta) \rangle}, which is minimized over the variational parameters \theta to yield an upper bound on E_0.[23] The expectation value \langle H \rangle can be expressed using the density operator \rho(\theta) = |\psi(\theta)\rangle \langle \psi(\theta) |, as \langle H \rangle = \mathrm{Tr} [\rho(\theta) H]. For practical computation on quantum hardware, the Hamiltonian is typically decomposed into a linear combination of Pauli operators: H = \sum_k c_k P_k, where c_k are real coefficients and P_k are tensor products of Pauli matrices. The expectation then becomes \langle H \rangle = \sum_k c_k \langle P_k \rangle, with each \langle P_k \rangle = \langle \psi(\theta) | P_k | \psi(\theta) \rangle obtained via quantum measurements.[23][24][2] At the variational minimum, the condition \frac{\partial E}{\partial \theta_j} = 0 holds for each parameter \theta_j. Differentiating the cost function yields \frac{\partial E}{\partial \theta_j} = 2 \mathrm{Re} \left[ \left\langle \frac{\partial \psi(\theta)}{\partial \theta_j} \Big| H - E(\theta) \Big| \psi(\theta) \right\rangle \right] = 0, which leverages the Hellmann-Feynman theorem in the context of parameter-shift rules or finite-difference approximations for gradient evaluation during classical optimization.[24][2] The variational principle guarantees convergence to the exact ground-state energy: E(\theta) \geq E_0 for any trial state, with equality achieved in the limit as the ansatz |\psi(\theta)\rangle spans the full Hilbert space of the system.[23][24]Algorithm Steps
The variational quantum eigensolver (VQE) operates through an iterative hybrid quantum-classical procedure to approximate the ground state energy of a given Hamiltonian. The steps are outlined below in a structured format, drawing from the original formulation and subsequent refinements for practical implementation.[10]- Encode the Hamiltonian into Pauli observables. The input Hamiltonian H, representing the quantum system of interest (e.g., from quantum chemistry or materials simulation), is transformed into a qubit-based representation. This involves mapping fermionic or other operators to a sum of Pauli strings: H = \sum_k c_k P_k, where c_k are coefficients and P_k are tensor products of Pauli matrices (I, X, Y, Z). Common mappings include the Jordan-Wigner or Bravyi-Kitaev transformations to ensure the encoding is compatible with qubit hardware. This step enables the expectation value \langle H \rangle to be computed additively from measurements of the individual P_k.[10]
- Initialize ansatz parameters \theta. A parameterized quantum circuit, or ansatz, is selected to generate trial states within a variational manifold. The parameters \theta (e.g., rotation angles in the circuit) are initialized, often starting from a mean-field solution such as the Hartree-Fock approximation to provide a physically motivated initial guess close to the ground state. Random initialization may also be used in some cases.[10]
- Quantum evaluation: Prepare |\psi(\theta)\rangle and measure \langle H \rangle via Pauli grouping. The trial state |\psi(\theta)\rangle is prepared on a quantum processor by executing the ansatz circuit with the current \theta. The energy E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle is estimated by measuring the expectation values \langle P_k \rangle for each Pauli term, typically grouped into commuting sets to minimize the number of distinct quantum circuits required (e.g., via qubit-wise or full commutativity partitioning). Multiple shots are performed per group to reduce statistical error. Measurement techniques are detailed in the Measurement Protocol section.[10]
- Classical update: Optimize \theta to minimize E(\theta); iterate until convergence. The estimated E(\theta) is passed to a classical computer, where an optimization routine (e.g., gradient-based or derivative-free methods) updates \theta to lower the energy. This quantum-classical feedback loop repeats, with the quantum evaluation providing the objective function evaluations for the optimizer.[10]