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The Variational Quantum Eigensolver: a review of methods and best practices

Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, Jonathan Tennyson

arXiv:2111.05176v3quant-ph

TL;DR

VQE is promising for quantum simulation, but its practical advantage remains uncertain because component costs may be too large. This review synthesizes VQE components and best practices, concluding that future applicability hinges on resolving open questions in measurement, scalability, optimization, and noise mitigation.

  • Problem

    VQE is investigated as a possible alternative to conventional computing, whose accuracy is constrained for central quantum chemistry and condensed-matter problems.

  • Method

    The review synthesizes VQE representations, optimization, error mitigation, and best practices while identifying open research questions about future applicability.

  • Results

    The review identifies measurement cost, parallelization, vanishing gradients, iteration scaling, and noise mitigation as central unresolved directions for VQE.

  • Takeaways & Limitations

    VQE’s future applicability depends on resolving open questions across its algorithmic components as quantum hardware scales and noise decreases.

  • Takeaways & Limitations

    Under current assumptions, studies conclude that VQE cannot outperform conventional methods for applications considered tractable, with observable sampling as the main bottleneck.

Abstract

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The variational quantum eigensolver (or VQE) uses the variational principle to compute the ground state energy of a Hamiltonian, a problem that is central to quantum chemistry and condensed matter physics. Conventional computing methods are constrained in their accuracy due to the computational limits. The VQE may be used to model complex wavefunctions in polynomial time, making it one of the most promising near-term applications for quantum computing. Finding a path to navigate the relevant literature has rapidly become an overwhelming task, with many methods promising to improve different parts of the algorithm. Despite strong theoretical underpinnings suggesting excellent scaling of individual VQE components, studies have pointed out that their various pre-factors could be too large to reach a quantum computing advantage over conventional methods. This review aims to provide an overview of the progress that has been made on the different parts of the algorithm. All the different components of the algorithm are reviewed in detail including representation of Hamiltonians and wavefunctions on a quantum computer, the optimization process, the post-processing mitigation of errors, and best practices are suggested. We identify four main areas of future research:(1) optimal measurement schemes for reduction of circuit repetitions; (2) large scale parallelization across many quantum computers;(3) ways to overcome the potential appearance of vanishing gradients in the optimization process, and how the number of iterations required for the optimization scales with system size; (4) the extent to which VQE suffers for quantum noise, and whether this noise can be mitigated. The answers to these open research questions will determine the routes for the VQE to achieve quantum advantage as the quantum computing hardware scales up and as the noise levels are reduced.

10 Conclusion and outlook · 1. Introduction

The review presents VQE as a promising NISQ approach for estimating ground-state energies while emphasizing that practical quantum advantage depends on controlling measurement, parallelization, optimization, and noise costs. It surveys the algorithm’s components and proposes best practices for molecular and spin-lattice systems.

  • 1. Introduction: VQE estimates an upper bound to a Hamiltonian’s ground-state energy, supporting electronic-structure calculations for molecules and materials.It was developed by Peruzzo et al. and formalized by McClean et al.
  • 1. Introduction: Quantum superposition can make equivalent wavefunction information exponentially costly to encode conventionally while requiring only linearly growing qubit resources.This motivates quantum simulation approaches such as VQE.
  • 1. Introduction: VQE models electronic-wavefunction physics and entanglement by applying a parameterized quantum circuit, whose ansatz structure and parameters determine the represented state.Circuit depth counts the number of consequential operations.
  • 1. Introduction: The review examines Hamiltonian and wavefunction representations, optimization, error mitigation, and best practices across ab initio molecular and spin-lattice systems.It aims to overview the VQE components and synthesize recommendations from the literature.
  • 1. Introduction: Polynomial measurement scaling can still become too costly for VQE, motivating compact Hamiltonian representations and joint measurement strategies.The number of required measurements may rapidly become too large for practical viability as system size increases.
  • 1. Introduction: VQE offers substantial parallelization potential, but efficient strategies and communication overheads have received relatively little study.Parallel execution is presented as an alternative way to address the measurement problem.
  • 1. Introduction: Barren plateaus can cause VQE parameter gradients to vanish exponentially with qubit number, ansatz expressibility, entanglement, or cost-function non-locality.This creates a central challenge for parameter optimization in variational quantum algorithms.
  • 1. Introduction: VQE’s resilience to quantum noise and tractable error mitigation remain unclear because mitigation costs can outweigh benefits, despite possible learning of systematic noise.The review also notes that its literature survey covers work published through the end of May 2022.

2. Overview of VQE

VQE estimates ground-state energies by optimizing a parametrized trial state, trading QPE’s circuit depth and qubit requirements for more measurements and an approximate ansatz. Its practical advantage remains uncertain because optimization, sampling, noise mitigation, and ansatz expressivity can impose substantial costs.

  • Ansatz and state preparation: An ansatz parametrizes a quantum circuit that prepares a trial wavefunction whose optimized state models the Hamiltonian ground state.The ansatz structure and parameters determine the trial state used for energy measurement.
  • Optimization: VQE’s practical cost depends on optimizer and landscape complexity, which is unfavorable because the algorithm is far from having a convex cost landscape.Even polynomial-time convergence results for convex functions do not directly establish tractable VQE optimization.
  • Optimization: Barren plateaus can make accurate gradient estimation require exponentially many measurements in certain system parameterizations.Thus, exact expectation-value and gradient assumptions can fail in quantum implementations.
  • Noise and error mitigation: Error mitigation can improve VQE accuracy on current quantum computers but may significantly increase resource requirements, leaving noise resilience unresolved.The passage identifies active error mitigation as likely unavoidable for relevant NISQ applications.
  • Classical benchmarks: Conventional high-accuracy wavefunction methods remain the benchmark for VQE because they can systematically approach chemical accuracy more efficiently than FCI.The applicability and errors of approximate conventional methods depend on system properties such as excitation rank, locality, or sparsity.
  • VQE–QPE tradeoff: VQE requires O(1/ε^2) shots and circuit depth O(1) in precision ε, whereas QPE requires O(1) repetitions and circuit depth O(1/ε).This tradeoff exchanges QPE’s depth and qubit demands for VQE’s repeated measurements and approximate state representation.

3. Hamiltonian Representation

Hamiltonian representation begins by formulating the ground-state eigenvalue problem and selecting compact, accurate bases for electronic wavefunctions. Different mappings and representations determine qubit requirements, Pauli-term counts, and practical suitability for VQE, including substantial costs for first quantization and distinct scaling for vibrational systems.

  • Problem formulation: The Hamiltonian representation frames VQE as finding the lowest eigenvalue of a Hermitian matrix, corresponding to an interacting system’s ground-state energy.For molecular systems, the target is the lowest-energy correlated electronic wavefunction.
  • Basis selection: Choosing a compact basis that accurately describes the system is critical because the required number of qubits scales with the number of basis functions.Minimal bases may need enlargement with polarization, diffuse, higher-energy, or higher-angular-momentum functions to treat correlation appropriately.
  • Electronic basis construction: Molecular orbitals are formed by linearly combining non-orthogonal atomic orbitals, typically through a mean-field calculation, before constructing many-body states.Correlated wavefunctions are expanded as linear combinations of occupied and unoccupied spin-orbital configurations encoded in qubit registers.
  • First quantization: First quantization produces O(n^4m^2) two-body operator terms that become n^2 Pauli strings, each acting on up to 2 log2(n) qubits.This scaling follows from combinations of four spin-orbitals and two electrons, with each spin-orbital represented by log2(n) qubits.
  • First quantization: First quantization has no known VQE or other NISQ studies and incurs significant ancilla and integral-computation costs despite unchanged overall scaling.The method requires computing all spatial integrals on the quantum computer.
  • Vibrational Hamiltonians: For V vibrational modes, direct mapping requires Vd qubits and O(V^k d^k) Pauli terms, whereas compact mapping requires Vlog2(d) qubits and O(V^k d^2k) terms.These counts apply when anharmonic terms up to order k are included.

4. Fermionic space to spin space transformations

Fermionic-to-spin mappings trade Pauli weight, operator count, qubit count, and noise behavior. Jordan-Wigner and parity have O(N) Pauli weight, while Bravyi-Kitaev and ternary-tree encodings reduce this scaling, with graph-based methods offering interaction-degree-dependent alternatives.

  • Standard mappings: Jordan-Wigner produces two Pauli strings per fermionic operator, so second-quantized Hamiltonians require O(n^4) strings.The mapping’s Z strings also give Pauli weight O(N), increasing the entangling-gate cost of fermionic ansätze.
  • Standard mappings: The Jiang et al. ternary-tree mapping achieves minimum possible average Pauli weight for fully connected fermionic Hamiltonians, with maximum weight log_3(2n + 1).Its operators map to Majorana modes along paths in a ternary tree, whose height determines the maximum weight.
  • Standard mappings: Jordan-Wigner and parity mappings have O(N) Pauli weight, whereas Bravyi-Kitaev scales O(log_2(N)) and optimal ternary-tree encoding scales O(log_3(2N)).All mappings use N = n qubits, equal to the number of fermionic modes.
  • Mapping trade-offs: Lower Pauli weight in Bravyi-Kitaev and ternary-tree mappings does not generally reduce readout errors after Pauli grouping, because groups often require measuring the full register in the Z basis.Their main advantage is constructing fermionic-operator ansätze rather than grouped joint measurements.
  • Mapping trade-offs: Bravyi-Kitaev UCC ansätze can show slightly higher sensitivity to quantum-noise channels than Jordan-Wigner despite lower gate depth.The proposed explanation is that Jordan-Wigner stores occupation numbers locally, limiting a single-qubit error to one orbital.
  • Graph-based encodings: Graph-based encodings make Pauli weight depend on interaction-graph degree: Superfast Encoding scales O(d), while SFBK scales O(2^d) weight and O(nd/2) qubits.Superfast Encoding uses qubits proportional to graph edges, and SFBK has been applied to Hubbard and ab initio molecular systems.

5. Efficient grouping and measuring strategies

Efficient VQE measurement strategies reduce shot requirements through optimized allocation, grouping, unitary decompositions, basis rotations, and classical shadows. These methods improve prefactors or scaling, but can introduce circuit-depth and computational costs.

  • Shot allocation and precision: Shot requirements scale as O(1/ϵ^2), while chemical-accuracy targets use ϵ = 1.6 mEH.Optimally distributing measurements among Pauli strings minimizes variance for a target precision.
  • Shot allocation and precision: Weight-pro rata measurement allocation, S_a ∝ |w_a|, generally outperforms uniform allocation for random states.Variations in Pauli-string weights tend to exceed variations in their variances, motivating weight-based allocation.
  • Shadow and basis-rotation methods: Classical shadows can make measurements scale logarithmically with the number of operators, while Basis Rotation Grouping reduces joint terms to linear system-size scaling.Classical shadows may have superior empirical scaling to Basis Rotation Grouping, whereas basis rotation requires O(N) basis changes and increases gate depth by O(N).
  • Commuting and unitary grouping: QWC grouping reduces the prefactor for measured Pauli terms by about three without changing asymptotic scaling.The grouping procedure is computationally cheaper than General Commuting grouping and saves shots through joint measurements.
  • Commuting and unitary grouping: General Commuting grouping reduces separate-term scaling from O(N^4) to O(N^3) for ab initio molecular Hamiltonians.Pauli strings commute when the number of index-wise noncommuting operators is even.
  • Commuting and unitary grouping: Unitary-group and anti-commuting strategies reduce Hamiltonian terms linearly, but unitary measurement can require O(q̃N^2) CNOTs.Appending unitaries enables joint measurement, with alternative entangling-gate scalings of O(N^2) under Jordan-Wigner and O(Nlog(N)) under Bravyi-Kitaev.

6. Ansatz selection and construction

Ansatz selection balances expressibility, trainability, circuit resources, and noise resilience. The review favors local, shallow, chemically structured ansätze over highly expressive hardware-efficient constructions when barren plateaus and scalability dominate.

  • Barren plateaus: If ansatz layers form 2-designs, gradient variance vanishes exponentially with system size, making sufficiently large gradients exponentially unlikely.Chebyshev’s inequality bounds the probability of a gradient exceeding an arbitrarily small threshold by an exponentially decreasing quantity; gradient-free optimizers can also be affected.
  • Barren plateaus: Higher expressibility lowers the gradient-variance bound and reduces trainability, although low-expressivity ansätze can also develop barren plateaus from size, initialization, or non-local costs.Trainability and expressibility are inversely related, so expressiveness alone does not determine susceptibility.
  • Barren-plateau resilience: Local encodings and ansätze with depth O(log(N)) measured using local observables are more resilient to barren plateaus than highly non-local alternatives.Bravyi-Kitaev, ternary tree, and Generalized Superfast encodings have lower Pauli weight than Jordan-Wigner, whose Pauli weight scales as O(N).
  • Noise-induced barren plateaus: Noise-induced barren plateaus can accelerate gradient decay and suppress expectation-value amplitudes independently of parameter initialization or cost-function locality.Consequently, initialization and locality strategies that help in noiseless settings may fail under noise.
  • Hardware-efficient ansätze: Hardware-efficient ansätze are resource-intensive and significantly limited by barren plateaus, making them mainly useful for proof-of-principle studies.A 12-qubit hardware-efficient ansatz example requires 198 parameters; some structures achieve reasonable Ising and XXZ accuracies with layers scaling at least linearly in system size.
  • Unitary coupled-cluster ansätze: Unitary coupled-cluster ansätze offer structured parameterizations with O(N^3) to O(kN^2) parameters and can be more resource-efficient than hardware-efficient ansätze.UCCGSD depth scales as O(N^3), while a single Trotter step can accurately describe simple molecular ground states; UCCGSD also improves results over UCCSD toward FCI for H2O.

7. Optimization strategies

VQE optimization iteratively learns ansatz parameters, but tractability is challenged by NP-hardness, measurement precision, sampling and gate noise, and barren plateaus. Analytical gradients, landscape approximations, gradient-free methods, collective optimization, and CVaR offer routes to improve convergence and robustness.

  • Optimization challenges: VQE optimization must learn an accurate wavefunction approximation within a tractable number of learning steps, although variational ansatz optimization is NP-hard.The NP-hardness result means some instances require exponentially many steps to reach an adequate solution.
  • Optimization challenges: Sampling and gate noise disturb the objective landscape, while finite shot numbers limit expectation-value precision and make optimization cost depend heavily on required precision.These effects can hinder convergence and potentially limit quantum advantage.
  • Barren plateaus: Barren plateaus can cause vanishing gradients, but natural-gradient methods have a lower gradient bound, analytical methods can jump toward optima, and CVaR emphasizes best samples without introducing local minima.Analytical methods incur increased measurement cost, whereas CVaR addresses gradient plateaus by emphasizing favorable observed outcomes.
  • Optimization strategies: Analytical landscape properties enable direct gradient evaluation and efficient landscape approximations, while parameter-shift implementation before fermionic encoding reduces required measurements.These techniques use prior structure to accelerate optimization and lower measurement overhead.
  • Optimizer comparisons: Natural-gradient, stochastic-approximation, Nelder-Mead, SPSA, and collective methods improve optimization in different settings, including faster convergence, limited measurements, or escape from local minima.Stochastic approximated natural gradient outperforms standard gradient descent but underperforms original natural gradient; Nelder-Mead is noise-sensitive, while SPSA often compares favorably.

8. Error mitigation for VQE

VQE error mitigation offers lower-resource alternatives to full error correction, but its methods can substantially increase measurement cost and variance. Key open issues include choosing effective mitigation strategies and determining when mitigation is necessary during optimization.

  • Motivation: NISQ error-mitigation techniques reduce noise in expectation-value estimates without the large resources required for error correction.They are described as critical for achieving the precision required in early VQE-based quantum chemistry implementations.
  • Symmetry verification: Symmetry verification mitigates errors by measuring a symmetry operator and post-selecting outcomes with the correct symmetry value.This filters errors that violate the symmetry regardless of the quantum computer’s error rate, although higher error rates increase the failure rate.
  • Extrapolation: Extrapolation-based mitigation multiplies circuit evaluations for each Pauli observable by the fitted-data size and increases observable variance by a factor γ.The section notes that this added cost can become prohibitively large.
  • Extrapolation: The exponential extrapolation model has significantly larger variance than the polynomial model but can predict exact measurement outcomes more accurately when enough sampling is affordable.The polynomial model is described as an approximation to the exponential model when noise is not very high.
  • Practical considerations: Error mitigation generally increases expectation-value variance, often exponentially with qubit count or the number of repeated ansatz layers.Its cost may be reduced by applying mitigation only after VQE optimization begins to converge, while fair comparisons across methods remain necessary.

9. Beyond the ground state of isolated molecules: Extensions of VQE

This section extends VQE beyond isolated-molecule ground states to excited-state and dynamical properties, and to hybrid approaches coupling correlated subspaces with wider environments. These extensions trade additional optimization, measurement, or classical resources for broader applicability.

  • Excited states: Excited-state computation is important for optical, transport, and reactive properties but is harder than ground-state computation because states may be farther from mean-field descriptions and optimization risks variational collapse.Excited-state methods must avoid collapsing to the ground state while treating more challenging wavefunctions.
  • Excited states: Quantum subspace expansion samples a low-dimensional approximate Hamiltonian on quantum computers and diagonalizes it classically to approximate higher-lying eigenvalues.The subspace dimension grows only as a low-order polynomial of system size, and equation-of-motion variants can avoid particularly deep circuits but require high-order reduced density matrices.
  • Excited states: Variational excited-state methods directly optimize ansätze with modified cost functions that preserve orthogonality, avoiding subspace-expansion limitations but generally requiring more quantum resources and a chosen ansatz.Examples include SSVQE, variational quantum deflation, discriminative VQE, and Variance VQE.
  • Dynamical correlation functions: Dynamical correlation functions encode excitation spectra across energy scales and can be represented in time or frequency domains, including through the single-particle Green’s function.These functions govern the linear-response behavior of quantum systems.
  • Quantum embedding: Embedding methods use VQE as a high-accuracy, scalable solver for correlated subspaces coupled self-consistently to wider environments, reducing qubit requirements while adding classical resources.Active-space optimization can require the two-body reduced density matrix, increasing measurement demands, while external-orbital correlation may be treated with low-order perturbation theory.

10. Conclusion and outlook

VQE is a promising near-term quantum-computing application, but its practical applicability remains uncertain because measurement costs, ansatz and optimizer choices, limited comparative studies, and other obstacles require further research.

  • Hamiltonian selection directly affects the qubit count and VQE’s capacity.
  • Measurement overhead is a major VQE drawback, motivating shot allocation, term truncation, and jointly measured operator methods.
  • The k-UpCCGSD ansatz and its extension combine excellent accuracy with scaling linear in the number of qubits.
  • Optimizer choice affects convergence and cost, while comparative evidence remains sparse; quantum natural gradient and RotoSolve have numerical support.
  • Comprehensive, comparative, applied studies remain rare, limiting assessment of VQE’s practical applicability and best practices.
  • VQE’s future value may depend on pairing it with related algorithms or QPE to accelerate computation, while overcoming obstacles to practical calculations.

A. Qubit encodings and Fenwick trees · B. Hadamard test

Fenwick trees provide a graphical construction for qubit encodings, translating Bravyi–Kitaev qubit sets into tree relationships and extending to parity, Jordan–Wigner, and new encodings. The Hadamard test estimates the real and imaginary parts of ⟨ψ|U|ψ⟩ from ancilla measurement probabilities.

  • A. Qubit encodings and Fenwick trees: Bravyi–Kitaev encodings can be represented as Fenwick trees constructed recursively, mirroring the change-of-basis matrix construction.The single-qubit encoding gives a trivial graph, while larger trees are built recursively.
  • A. Qubit encodings and Fenwick trees: In the Fenwick-tree representation, the update set U(j) contains the ancestors of vertex j.This identifies update operations directly from the tree structure.
  • A. Qubit encodings and Fenwick trees: The flip set F(j) consists of vertex j’s children, while the remainder set R(j) contains children of ancestors with values less than j.These correspondences translate the Bravyi–Kitaev qubit-set definitions into simple graph relationships.
  • A. Qubit encodings and Fenwick trees: For non-power-of-two qubit counts, the construction uses the next power-of-two Fenwick tree and discards qubits greater than or equal to n.The resulting sets can be read from the tree and used to represent creation and annihilation operators.
  • A. Qubit encodings and Fenwick trees: Parity encoding corresponds to a linear Fenwick graph, whereas Jordan–Wigner encoding corresponds to a totally disconnected graph.Disconnected Fenwick trees also generalize the construction to other fermionic encodings.
  • B. Hadamard test: The Hadamard test uses an ancilla and controlled-U operation to estimate the amplitude ⟨ψ|U|ψ⟩ for an initial state |ψ⟩.The procedure measures the real part in one circuit and the imaginary part in a second circuit.
  • B. Hadamard test: For the real-part circuit, the probabilities of ancilla outcomes 0 and 1 are (1+Re⟨ψ|U|ψ⟩)/2 and (1−Re⟨ψ|U|ψ⟩)/2, respectively.Their difference equals Re⟨ψ|U|ψ⟩.
  • B. Hadamard test: In the second circuit, the difference between ancilla outcome probabilities 0 and 1 equals Im⟨ψ|U|ψ⟩.Thus, the two circuit variants recover both components of the amplitude.

C. Error mitigation appendix … C.1.5. Damping error

The appendix surveys common quantum-computing noise models relevant to error mitigation, covering relaxation, over-rotation, depolarizing noise, dephasing, and damping. It characterizes their physical effects, parameters, and implications for detecting or mitigating errors.

  • C.1. Common noise models: Common noise models are introduced as representative models used in the quantum-computing literature, with further details deferred to Ref..The appendix frames these models as common examples rather than an exhaustive treatment.
  • C.1.1. Relaxation rates: T1 and T2 relaxation times quantify single-qubit state errors caused by thermalization with the environment.They are described as figures of merit for quantum-computer noise robustness.
  • C.1.1. Relaxation rates: T1 and T2 denote longitudinal and transverse relaxation rates that can be measured experimentally and are widely available for quantum computers.The text also identifies a0 as the thermal-equilibrium-state parameter, usually a0 = 1 for superconducting qubits.
  • C.1.2. Over-rotation: Over-rotation error arises from miscalibrated physical pulses, changing a Pauli rotation R_i(θ) into R_i(θ + δ).The angle deviation δ may be systematic or stochastic and can vary across quantum computers.
  • C.1.3. Depolarizing noise model: The depolarizing model N_depolar changes an n-qubit state ρ through noise whose strength is characterized by ε.The model uses the identity matrix on n qubits in its state transformation.
  • C.1.4. Dephasing noise model: Dephasing noise describes the disappearance of density-matrix off-diagonal elements and is closely related to the T2 relaxation rate.Its single-qubit action depends on the initial-state parameters a and b and noise-strength parameter ε.
  • C.1.4. Dephasing noise model: Dephasing does not affect the diagonal part of a single-qubit state, so it commutes with most symmetry operators and cannot be detected by symmetry verification.This limitation is stated for the symmetry-verification technique described in Sec. 8.1.
  • C.1.5. Damping error: Damping error decreases the population of |1⟩ in a single-qubit state and is closely related to the T1 relaxation time.The model is represented by Kraus operators, with ε characterizing its strength and determining its action on the initial state ρ.

C.2. Example for probabilistic error cancellation · C.3. Implementation of exponential error suppression

The section illustrates probabilistic error cancellation for a one-qubit depolarizing channel and describes the circuit construction used to cancel noise. It also introduces implementation of exponential error suppression through numerator evaluation, with the denominator treated as an identity-observable special case.

  • C.2. Example for probabilistic error cancellation: The example targets depolarizing noise on a one-qubit quantum computer using probabilistic error cancellation.The depolarizing channel is parameterized by the noise strength and expressed using Pauli matrices.
  • C.2. Example for probabilistic error cancellation: The depolarizing-channel representation includes the quasi-probability coefficient q_dep,2 = −p/(2p + 4).This coefficient appears in the decomposition used for probabilistic error cancellation.
  • C.2. Example for probabilistic error cancellation: Three additional circuits append X, Y, or Z after the ideal unitary U, alongside the original circuit U.Measurement outcomes from these four circuits are combined according to the quasi-probability weights.
  • C.2. Example for probabilistic error cancellation: Using the weighted outcomes from the original and Pauli-appended circuits cancels the noise effect of N_1 through PEC.The construction represents U acting on ρ with appended Pauli gates and applies the weights from Eq. (323).
  • C.3. Implementation of exponential error suppression: Implementation of exponential error suppression is discussed through methods for computing the numerator in Eq. (304) on quantum computers.The treatment focuses mostly on the numerator for simplicity.
  • C.3. Implementation of exponential error suppression: The denominator is treated as a special case of the numerator when the observable O is the identity.This reduces the implementation discussion to numerator computation.

C.3.1. Ancilla assisted method

The ancilla-assisted method estimates Tr[ρ^K O] by using K copies of a state, permuting their indices, and measuring an overlap with an additional ancilla qubit. The permutation can be implemented with a derangement circuit and evaluated using a modified Hadamard test, with variants reducing copy requirements and tolerating some noise.

  • Method: The method uses extra ancilla qubits and K copies of the same state ρ⊗K to measure O on one system through an overlap with a permuted state.The target observable is evaluated by comparing the original and permuted states.
  • Permutation structure: A permutation s of the K copy indices enforces the delta functions, yielding Tr[ρ^K O] for the statistical mixture ρ⊗K.The permutation is a K-cycle derangement with distinct indices.
  • Circuit implementation: Given s, a unitary circuit D_K implements the required mapping between the original and permuted states.For the example permutation, D_K uses CNOTs on qubit pairs (1, 2), …, (K−1, K), and (K, 1).
  • Measurement: The overlap can be measured with a modified Hadamard test after constructing the permuted state.The procedure is illustrated by the circuit in Fig. 17a.
  • Improvements and noise: A deep-circuit variant performs derangement on a single copy, avoiding K copies, while noisy D_K remains valid if it preserves the orthogonal relations.The long-range derangement circuit may be prone to errors on actual quantum computers.

C.3.2. Diagonalization method

The diagonalization method reformulates derangement as a matrix and implements unitary diagonalizations of the derangement and observable-transformed operators to estimate the numerator and denominator. Because the derangement permutes copies, these diagonalizations can be reduced to local operations, and grouping can simplify the resulting measurements.

  • Method: The method requires no additional ancilla qubit or long-range controlled-D_K operation, using only local B_i operations connecting corresponding qubits across copies.It reformulates derangement as a matrix S and implements unitary versions of S and O(1)S.
  • Diagonalization: When S and O(1)S can be unitarily diagonalized, applying their diagonalizing unitaries and measuring the K copies in the computation basis yields the numerator and denominator of Eq. (304).The diagonal forms are represented by the matrices Λ_S and Λ_OS.
  • Factorization: Because S permutes the K copies, it factorizes into tensor products of operators S_i that independently permute corresponding qubits across copies.For K = 2, each S_i interchanges qubits occupying the same position in the two copies.
  • Local simplification: If the observable acts on only a few qubits, diagonalizing O(1)S reduces to diagonalizing the corresponding S_i operators and an additional operator on the observable’s qubit subset.For one-qubit observables, a symmetrized observable can make the diagonalization unitaries coincide and factorize into local unitaries.
  • Measurement: Grouping methods convert complex multiqubit measurements into simple one-qubit measurements, making the implementation appealing because it requires only local qubit connections.The simultaneous diagonalization example prepares two copies with the same ansatz circuit and measures after applying diagonalizing gates to corresponding qubit pairs.

C.3.3. Dual state purification method

The dual state purification method applies a state-preparation circuit and its inverse, with post-selection used in one implementation to support measurement. The method extends to arbitrary observables by decomposing them into linear sums of Pauli observables.

  • Circuit implementation: The state-preparation circuit U is applied twice, first as U and then as its inverse U†.The circuit prepares a state ρ from |0⟩ before applying U†.
  • Circuit implementation: One implementation performs a mid-circuit measurement and post-selects experiments whose final measurements on all qubits yield 0.This example measures Pauli-Z on the first qubit, denoted Z1.
  • Observable extension: The method can measure any observable O by expanding it into a linear sum of Pauli observables.An explicit extension is available for observables O that square to the identity, including Pauli observables.

C.3.4. Shadow tomography based method

The shadow tomography-based method estimates the numerator Tr(휌K̂O) using randomized measurements, with a more complex variant incorporating error-correction codes.

  • The method was proposed in Refs. [748] [749] and uses shadow tomography to estimate the quantity in Eq. (304).
  • More simply, the approach estimates the numerator Tr(휌K̂O) in Eq. (325).
  • The procedure applies a random unitary U from a pool to 휌 and measures the resulting state in the computational basis.
  • The method in Ref. [749] is more complicated because it involves error-correction codes but shares the same shadow-tomography-based strategy.
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