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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

Whether VQE’s costs permit an advantage over conventional methods remains unresolved. This review synthesizes the algorithm’s components and best practices, and reports k-UpCCGSD as offering excellent accuracy with linear scaling and a strong cost-to-accuracy trade-off.

  • Problem

    Whether VQE’s component costs permit it to outperform conventional methods remains uncertain, with measurement costs identified as a major bottleneck.

    Source: §1 · p. 4, §2.3 · p. 12

  • Method

    The authors review VQE components, suggest literature-based best practices for molecular and spin-lattice systems, and identify open research questions.

    Source: §1 · pp. 5–6

  • Results

    k-UpCCGSD is reported to achieve excellent accuracy with linear scaling, arguably offering the best cost-to-accuracy trade-off among ansätze proposed at the time of writing.

    Source: §6.4 · p. 88

  • Takeaways & Limitations

    Practical variational algorithms require optimization strategies that find a well-approximated solution within an acceptable number of iterations.

    Source: §7 · p. 88

    Under stated assumptions and for applications considered tractable, prior cost studies conclude VQE cannot outperform conventional methods, with measurement burden the main noiseless bottleneck.

    Source: §2.3 · p. 12

Abstract

from arXiv · show

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.

1. Introduction

VQE seeks ground-state energy estimates for molecular and material systems where accurate conventional calculations become intractable, using quantum circuits to model wavefunctions. This review surveys the algorithm’s components and best practices while identifying unresolved challenges to its practical viability.

Motivation

  • Accurate conventional calculations become intractable as electron correlations demand exponential scaling, motivating VQE’s potential to model quantum wavefunctions beyond classical capabilities.The review notes applications across electronic structure and other quantum-mechanical problems, including nuclear physics and spectroscopy.Source: §1 · pp. 4–5

VQE method

  • VQE variationally optimizes a parameterized quantum circuit to estimate a Hamiltonian’s ground-state energy, with compact ansätze balancing circuit cost against wavefunction accuracy.The variational principle keeps the trial energy at or above the exact ground-state energy; shallower circuits generally cover a smaller span of possible wavefunctions.Source: §1 · p. 5

Review scope

  • The review surveys Hamiltonian representation, operator mapping, ansätze, optimization, measurements, and error mitigation, and proposes best practices for ab initio molecular systems and spin-lattice models.It also identifies four interrelated research hurdles affecting VQE’s future applicability.Source: §1 · pp. 5–6

Open challenges

  • Measurement counts may scale polynomially yet become impractically large, making compact Hamiltonian representations and joint measurements of commuting observables important research directions.The review also highlights efficient parallelization across quantum computers, including strategies and communication overheads, as comparatively underexplored.Source: §1 · p. 6
  • Barren plateaus can make VQE parameter gradients vanish exponentially with qubit count, ansatz expressibility, entanglement, or cost-function non-locality, while optimization scaling remains poorly understood.Local minima and non-convex landscape features may also undermine optimization; it remains unclear whether proposed methods can contain barren plateaus in VQE.Source: §1 · p. 6
  • VQE’s resilience to quantum noise and the tractability of mitigating it remain unclear, despite evidence that variational algorithms can learn away some systematic noise.Some mitigation methods may cost more than their benefits, so the review calls for deeper study of noise impacts and error predictability.Source: §1 · p. 6

2. Overview of VQE

This section surveys how VQE estimates ground-state energies through a quantum-classical optimization pipeline and situates it among exact and approximate wavefunction methods and QPE. It reviews system-dependent practices for molecular and lattice systems, examining resource costs, accuracy, scalability, hardware constraints, parallelization, and the division of work between quantum and conventional computers.

2. Overview of VQE

  • This section sets out VQE's technical foundations and its position relative to conventional electronic-structure and other quantum methods, alongside best practices and resource considerations.Source: §2 · p. 7

2.1. A formal definition of the VQE

  • VQE minimizes a parameterized trial state's Hamiltonian expectation, an upper bound on ground-state energy, to approximate the lowest eigenstate at the desired precision.A quantum device prepares the state and measures Pauli-string expectations; a conventional computer forms their weighted sum and performs the minimization.Source: §2.1 · pp. 7–8

2.2. The VQE pipeline

  • The VQE pipeline runs from Hamiltonian construction and encoding through measurement, ansatz preparation, parameter optimization, and error mitigation.The basis and representation affect computational size and accuracy; ansätze must balance expressibility and trainability, while measurement, optimization, and mitigation choices affect overall cost.Source: §2.2 · pp. 8–10

2.3. Advantage argument, assumptions, and limitations of the VQE

  • Practical quantum advantage requires VQE to match or exceed conventional accuracy faster on a physically relevant system, with Hamiltonian-model error smaller than solution error.A proposed estimate places the basis size for tangible quantum advantage at 19–34 molecular orbitals, or twice as many spin orbitals and qubits [83].Source: §2.3 · pp. 11–12
  • Large measurement costs and NP-hard optimization may prevent VQE from outperforming conventional methods, leaving heuristic convergence in a polynomial number of iterations an open question.Cost studies identify observable sampling as a bottleneck under their assumptions, while parallelization and improved measurement strategies are possible responses.Source: §2.3 · p. 12
  • Barren plateaus can make gradient measurements scale exponentially, while the costs and scalability of error mitigation and VQE's noise resilience remain uncertain.Mitigation has improved accuracy on current devices but may become exponentially more costly with deeper ansätze; small-scale noise resilience has not been established for larger experiments, and exact recovery is expected to require exponentially many variational parameters [161].Source: §2.3 · pp. 12–13

2.4. VQE and conventional computational chemistry

  • VQE draws on conventional quantum chemistry for basis orbitals, and its likely short- to medium-term competitors are accurate wavefunction methods rather than DFT.DFT offers a competitive cost-accuracy trade-off for large systems but lacks systematic improvability, so the review does not treat it as a direct competitor.Source: §2.4 · p. 13

2.4.1. Full Configuration Interaction

  • Full configuration interaction is the exact-state benchmark for a specified Hamiltonian and basis, but is usually intractable, with practical ab initio limits around 18 orbitals.Its linear parameterization spans all configurations and turns Ritz-functional minimization into Hamiltonian diagonalization; higher eigenvalues give exact excited states within that basis.Source: §2.4.1 · pp. 13–14
  • Chemical accuracy means results within 1.6 mEH of experimental reaction enthalpies, whereas chemical precision concerns solving the chosen model regardless of its approximations.Other properties, including spectroscopy, can require two to three orders of magnitude greater accuracy.Source: §2.4.1 · p. 14

2.4.2. Efficient approximate wavefunction parameterizations for conventional computation

  • Conventional approximate wavefunction methods are the demanding benchmark for VQE: they can approach FCI accuracy at larger scales, but their accuracy depends on system properties and ansatz structure.Truncated CI can approximate FCI well for small systems but may have size-intensive energies; DMRG, selected CI, and coupled-cluster methods exploit different structures. The review says VQE’s errors and domains of applicability remain insufficiently classified.Source: §2.4.2 · pp. 14–15

2.5.1. Overview of the quantum phase estimation

  • QPE uses controlled Hamiltonian-evolution unitaries to encode eigenvalue phases, then inverse QFT and ancilla measurement recover an eigenvalue with probability tied to input-state overlap.The ancilla count sets phase precision; measuring the ancilla register yields the phase-associated eigenvalue, with outcome probability determined by the input state's eigenbasis amplitude.Source: §2.5.1 · p. 17

2.5.2. Discussion and comparison

  • QPE’s ancilla count scales as O(1/log2(ε)) and controlled-unitary count as O(1/ε), but imperfect input-state fidelity can make success exponentially costly.Large-scale controlled unitaries and inverse QFT are not reliably implementable on NISQ devices; resource estimates for target chemistry problems remain demanding.Source: §2.5.2 · pp. 17–18
  • VQE trades QPE’s deeper circuits and fewer repetitions for more measurements and shallow circuits; α-VQE interpolates between their precision scalings, making them potentially complementary.For precision ε, the review gives QPE O(1) repetitions and depth O(1/ε), versus VQE O(1/ε^2) shots and depth O(1). α-VQE uses depth O(1/ε^α) and samples O(1/ε^(2(1−α))).Source: §2.5.2 · p. 18

2.6. Our suggested best practices for VQE and their scaling assessment

  • Best practices are conditional on system and hardware: ab initio molecular systems and lattice models differ in encodings, measurement strategies, and ansätze, while recommended scalings remain uncertain.The review limits its recommendations to these two families because research on alternative applications is sparse.Source: §2.6 · pp. 18–19, Table 1 · p. 19

2.6.1. Best practices for ab initio electronic structure of molecular systems

  • For ab initio systems, the review favors compact molecular-orbital bases and symmetry tapering; it identifies ternary-tree mapping as promising but says its noise-resilience benefit is unclear.Second quantization is generally preferred despite first quantization’s possible qubit advantage for few-electron, large-basis systems; its VQE efficiency has not been rigorously studied.Source: §2.6.1 · pp. 20–21
  • For molecular Hamiltonian measurements, decomposed-interaction grouping offers O(N) groups with O(N) basis-rotation depth, while classical shadows remain promising but need further comparison.The review favors predictable-cost Basis Rotation Group methods cautiously; general-commutativity grouping can require O(N^2) circuit depth, and group count alone does not determine shots needed for precision.Source: §2.6.1 · p. 21
  • For ab initio molecules, k-UpCCGSD offers linear circuit scaling, while Rotosolve or Fraxis are proposed optimizers despite costs and unresolved scaling of convergence.The ansatz has O(kN^2) parameters and yields an estimated O(k^2N^4/ε^2) per iteration without parallelization, versus O(kN) circuit depth with full parallelization; near convergence, smaller gradients can require more shots. Joint measurement bases may also introduce covariances that increase measurement costs.Source: §2.6.1 · pp. 21–22

2.6.2. Best practices for lattice models

  • For correlated lattice models, the Hamiltonian variational ansatz (HVA) is the safest general choice, exploiting compact lattice structure and showing resilience to barren plateaus [150].Its parameters and depth scale with the number of commutative Pauli groups, while estimated gradient computation for regular nearest-neighbor hypercubic lattices scales as O((kn^D)^3/ε^2); whether grouping reduces operator-count scaling remains unresolved. Encoding choices trade Pauli weight against qubit count, and QWC grouping is recommended pending further research.Source: §2.6.2 · pp. 23–24

2.7.1. Cost and runtime estimates for VQE

  • A single set of qubits appears orders of magnitude too costly for the estimated Cr2 VQE, making parallelization central to viability.The estimate assumes a 52-qubit active space, k = 1, a circuit depth of about 27,000 timesteps, and 1,000,000 shots; the last iterations dominate cost. Earlier estimates range from 1.9 days for methane to 71 days for ethanol, but do not account for optimization complexity, barren plateaus, or quantum noise.Source: §2.7.1 · p. 25

2.7.2. Parallelization potential of the VQE

  • Full VQE parallelization could trade runtime for hardware, but its O(pN^4/ε^2) compute cores may be prohibitive and introduce communication overhead and additional noise.Additional noise may arise from crosstalk in multithreaded devices or from differing systematic biases across quantum computers, which can undermine variational noise resilience.Source: §2.7.2 · pp. 25–26

2.7.3. Distribution of resources between quantum and conventional computation

  • Future VQE applications may need a more deliberate split between quantum and conventional computation, beyond quantum sampling and classical parameter updates.Proposals include partially solving many-body Hamiltonians classically, classical optimization for local-interaction states [305], and Hamiltonian dressing through ClusterVQE [306] to absorb noise-prone or costly ansatz components.Source: §2.7.3 · p. 26

3. Hamiltonian Representation

Hamiltonian representation determines how molecular, lattice, vibrational, and periodic-system problems are encoded for VQE, with representation choices affecting the computational resources required. The section explains electronic Hamiltonian construction, wavefunction bases, and first- and second-quantized forms.

3.1.1. The ab initio molecular Hamiltonian

  • Under the non-relativistic Born–Oppenheimer approximation, the molecular electronic Hamiltonian depends parametrically on nuclear positions, up to nuclear–nuclear repulsion.It represents the energy operator for a molecular system specified by its atomic composition and nuclear geometry, with the goal of finding the ground-state energy.Source: §3.1.1 · p. 27

3.1.2. Construction of the wavefunction

  • Electronic wavefunction bases must balance compactness and accuracy because qubit requirements scale with the number of basis functions.Minimal STO basis functions are weighted Gaussian sums, while correlated calculations may require larger polarized and diffuse bases; frozen natural orbitals and active-space approaches offer ways to contract or optimize molecular orbitals.Source: §3.1.2 · p. 28

3.1.3. Hamiltonian quantization

  • First quantization uses N = m log2(n) qubits but produces O(n^6m^2) Pauli strings; despite promise for large basis sets, it is currently costly for NISQ VQE.Maintaining antisymmetry also uses an ancilla register of O(m log2(n)) qubits and adds circuit depth; the review notes no published studies of first quantization in VQE or other NISQ algorithms. Second-quantized two-body reduced density matrices retain the information needed for observables, including total energy.Source: §3.1.3 · pp. 30–31

3.2.1. Lattice Hamiltonians

  • Lattice Hamiltonians describe particles on discrete sites and model condensed-matter properties, with examples including Hubbard, Anderson impurity, and Heisenberg models.The Anderson impurity model separates local impurity interactions, impurity–bath hopping, and a non-interacting bath.Source: §3.2.1 · p. 32

3.2.2. Vibrational Hamiltonian model

  • Vibrational Hamiltonians can use direct or compact mappings, trading qubit count against Pauli-term count as anharmonic order and mode count increase.For V modes and d eigenstates per mode, direct mapping uses Vd qubits and O(V^k d^k) terms, while compact mapping uses V log2(d) qubits and O(V^k d^2k) terms.Source: §3.2.2 · p. 33

3.2.3. Periodic systems

  • Periodic-system Hamiltonians extend molecular electronic structure with boundary conditions, and their integrals obey crystal-momentum constraints.The one-electron integrals are diagonal in k, while the two-electron integrals satisfy k_p + k_q − k_r − k_s = G, with G a reciprocal lattice vector.Source: §3.2.3 · p. 33

4. Fermionic space to spin space transformations

This section reviews mappings of fermionic operators to qubit or spin representations, including generalized and lattice-tailored encodings, and their role in measurable Pauli operators and ansatz circuits. It compares trade-offs in qubit count, Pauli weight, number of Pauli strings, connectivity, and noise handling, and describes symmetry-based methods for reducing qubit requirements.

4.1. Generalized encodings

  • Generalized encodings use n qubits and produce O(n^4) operators for molecular Hamiltonians, but generally map local fermionic terms to non-local Pauli strings.They are suited to ab initio molecular systems; geometry-specific mappings instead target low Pauli weight for low-degree lattice models, sometimes using additional qubits.Source: §4.1 · p. 35, Table 4 · p. 35, §4 · p. 35

4.1.1. The Jordan-Wigner encoding

  • Jordan–Wigner’s Z strings preserve fermionic anticommutation but give Pauli weight O(N); spin-operator-level ansatz parameters can cap it at four and reduce entangling gates.The number of Pauli strings scales as O(N^4) for a second-quantized Hamiltonian.Source: §4.1.1 · p. 36

4.1.2. The parity encoding

  • Parity encoding stores cumulative orbital parity and flips later qubits when an orbital changes, yielding Pauli weight O(N) and operator scaling O(N^4).A qubit records whether the number of occupied orbitals up to its index is even or odd.Source: §4.1.2 · pp. 36–37

4.1.3. The Bravyi-Kitaev encoding

  • Bravyi–Kitaev combines parity and occupation information to reduce maximum Pauli weight to O(log2(N)), while producing O(N^4) spin-operator terms.Its recursive construction bounds the size of the relevant qubit sets by O(log2(N)).Source: §4.1.3 · pp. 37–38

4.1.4. Optimal general encoding based on ternary trees

  • The ternary-tree encoding achieves maximum Pauli weight O(log3(2n + 1)), optimal for fully connected fermionic modes, while using n qubits and O(n^4) Pauli operators.It maps Majorana operators to tree-path operators; the maximum weight equals the tree height.Source: §4.1.4 · pp. 39–40
  • Jiang et al.’s ternary-tree mapping uses Majorana operators and achieves the minimum average Pauli weight possible for fully connected fermionic modes [256].Its maximum Pauli weight is the tree height, log3(2n + 1).Source: §4.1.4 · pp. 39–40

4.1.5. Discussion on generalized encondings

  • Fenwick-tree constructions can express generalized encodings and help optimize Bravyi–Kitaev for two-dimensional lattice models.Steudtner et al. also propose symmetry-based encodings that reduce qubits at the cost of additional entangling gates.Source: §4.1.5 · pp. 40–41
  • Jordan–Wigner and Parity have O(N) Pauli weight, versus O(log2(N)) for Bravyi–Kitaev and O(log3(2N)) for the optimal ternary tree; grouping can erase readout benefits, while low weight may aid fermionic ansätze.The mappings use the same number of qubits and can benefit equally from symmetry-based qubit reduction. Low Pauli weight may increase entangling-gate requirements on limited-connectivity registers; Bravyi–Kitaev VQE ansätze may be more noise-sensitive than Jordan–Wigner despite lower depth, possibly because errors affect multiple occupation numbers.Source: §4.1.5 · p. 41

4.2. Lattice model tailored encoding

  • Lattice-tailored mappings address non-local operations from full-connectivity encodings; they trade additional qubits for lower Pauli weights, can enable some error correction, and produce operator counts scaling linearly with lattice edges.For a regular D-dimensional lattice, the edge count is capped below O(nD), with auxiliary sites adding a prefactor but not changing overall scaling.Source: §4.2 · p. 41

4.2.1. Auxiliary fermion schemes

  • Auxiliary-fermion schemes cancel Jordan–Wigner Z strings on uncovered lattice edges, capping nearest-neighbor hopping and Coulomb Pauli weights at 4 and 2, respectively, but doubling the qubit requirement.In one dimension, Jordan–Wigner already preserves locality; for an h×v rectangular lattice, naive Jordan–Wigner can reach maximum Pauli weight O(2v), compared with 4 under the auxiliary scheme.Source: §4.2.1 · pp. 42–44

4.2.2. Superfast Encoding / Loop Stabilizer Encodings

  • Superfast Bravyi–Kitaev places qubits on interaction-graph edges, giving Pauli weight O(d) and qubit count proportional to edges; for degree d, these scale as O(2d) and O(nd/2), respectively.Its loop stabilizers define a codespace encoding n−1 logical qubits into N physical qubits and restrict the simulation to the even-parity sector.Source: §4.2.2 · pp. 44, 46
  • Generalized Superfast Encoding retains SFBK’s qubit scaling but lowers edge-operator Pauli weight to O(log(d)); its codespace can detect single-qubit errors when vertex degree is at least 6.Its loop-stabilizer codespace provides error correction, but the error-threshold tolerance has not been estimated.Source: §4.2.2 · pp. 46–47
  • The compact encoding uses at most 1.5 qubits per site or fermionic mode and avoids the parity-condition weight requirement, but even-face stabilizers impose constraints and the method detects many single-qubit errors.Extensions to other uniform lattices of degree up to 4 and cubic lattices are reported in [261].Source: §4.2.2 · pp. 47–48

4.2.3. Discussion on lattice tailored encodings

  • The compact encoding offers the best combination of low qubit count and Pauli weight up to three dimensions, while GSE has the lowest Pauli weight but scales qubit count with graph degree and system size.Extending the compact encoding to higher dimensions requires further research.Source: §4.2.3 · p. 48
  • Encoding comparisons remain unsettled: SFBK has lower gate depth than Jordan-Wigner but higher than Bravyi-Kitaev, while Jordan-Wigner can have lower gate count than SFBK under some Hamiltonian conditions.Meaningful comparisons require equivalent error-mitigation resources, and whether lattice mappings’ error-mitigating or correcting properties are useful under realistic noise remains unclear.Source: §4.2.3 · p. 48

4.3. Reducing qubit requirements

  • Tapering qubits associated with conserved Z2 symmetries halves the Hilbert-space dimension per removed qubit, and molecular point-group symmetries can remove at least as many qubits as the earlier method.The point-group approach removes 5 qubits for CO2 on 30 qubits, C2H2 on 24 qubits, and BeH2 on 14 qubits; the symmetry-finding procedure is limited to tensor products of Pauli operators.Source: §4.3 · p. 49

5. Efficient grouping and measuring strategies

This section reviews strategies for reducing VQE measurement costs, including shot allocation, inference, and grouping Hamiltonian terms. It considers trade-offs in circuit overhead, covariance, and grouping complexity, and discusses measurement approaches for molecular Hamiltonians.

5.1.1. Overall scaling of measurements:

  • Shot requirements for target error ε scale as O(1/ε²), but VQE chemical accuracy is ε=1.6 mEH and does not predict gradient-estimation cost: barren-plateau gradient differences may be orders smaller, requiring many more measurements.For Fe2S2 ferredoxin in STO-3G with N=112 spin-orbitals, Ref. conservatively estimates O(10^19) shots.Source: §5.1.1 · p. 50

5.1.2. Measurement weighting

  • Term truncation removes terms with contributions significantly below error tolerance ε; it reduces quantum-chemistry calculation cost with negligible accuracy impact when bias is below the mean-square-error reduction from measurements on remaining terms.Source: §5.1.2 · pp. 51–52

5.2.1. Inference methods

  • Quantum Overlapping Tomography uses overlapping measurements to reduce shots for k-qubit reduced density matrices, while Bayesian Hamiltonian Learning estimates P strings with O(3^k log(P)/epsilon^2) parallel shots.Both approaches target Hamiltonian characterization and may not be directly optimized for VQE ground-state estimation; QOT also may not exploit existing Hamiltonian structure [387].Source: §5.2.1 · pp. 52–53
  • Classical shadows can make measurement requirements logarithmic in the number of operators, and numerical studies suggest better empirical scaling than Basis Rotation Grouping.Weighted measurements can address exponential scaling for non-local observables, and low-weight derandomization selects influential Pauli strings deterministically; shadows have also been studied for noise resilience and barren-plateau mitigation [121] [392] [393] [394] [395] [402] [403].Source: §5.2.1 · pp. 53–54
  • Neural-network tomography uses a restricted Boltzmann machine to learn an ansatz-state approximation for estimating observables, restricting training measurements to Hamiltonian Pauli strings to avoid exponential sums.The original formulation’s unrestricted measurement-basis and outcome sets scale exponentially, motivating finite measurement sets [120].Source: §5.2.1 · pp. 54–55

5.2.2. Hamiltonian partitioning based on commutativity

  • Qubit-wise commutativity grouping enables joint measurements in a tensor-product basis, reducing the Pauli-term measurement prefactor by about three without changing asymptotic scaling.QWC partitioning is computationally cheaper than General Commutativity partitioning [117].Source: §5.2.2 · p. 55
  • For ab initio Hamiltonians, general-commutativity grouping reduces the number of terms to measure from O(N^4) to O(N^3), but requires entangling basis rotations whose gate cost depends on the ansatz.Qubit-wise commuting groups need only depth-1 basis rotations and save shots without changing overall scaling; the added cost of general-commutativity rotations can be significant for linearly scaling ansätze.Source: §5.2.2 · pp. 55–56

5.2.3. Unitary partitioning of the Hamiltonian:

  • Anti-commuting partitions express the Hamiltonian as a weighted sum of unitary operators and can linearly reduce its number of terms.Overlap-based measurement at least doubles ansatz depth and has CNOT scaling O(qN^2); an alternative basis-rotation approach has entangling-gate scaling O(N^2) with Jordan–Wigner and O(N log N) with Bravyi–Kitaev.Source: §5.2.3 · pp. 57–59

5.2.4. Decomposed interactions for efficient joint measurements:

  • Basis Rotation Grouping decomposes two-body interactions to reduce jointly measured terms to linear scaling, with O(N) added gate depth and improved noise resilience for high-Pauli-weight mappings.Although FRO can use fewer partitions than Basis Rotation Grouping, accounting for covariance found GC with Sorted Insertion best in most tested systems; VGFRO, GFRO, and basis rotation performed similarly and better than FRO [262].Source: §5.2.4 · pp. 59–60

5.2.5. Grouping heuristics

  • Grouping heuristics map Pauli-string grouping to the NP-hard minimum clique cover problem, and Sorted Insertion targets measurement count rather than group count.Sorted Insertion achieved the highest shot reduction for a given precision in several systems up to 16 qubits; conventional heuristics can scale as O(N^k4), with LDFC at O(N^8) for second-quantized Hamiltonians.Source: §5.2.5 · pp. 61–62

5.3. Discussion on measurement strategies and grouping methods

  • The best grouping method should be judged by measurements needed for target precision, not just group count, because joint measurements can introduce covariance and basis-rotation circuits can add quantum noise.Grouping heuristics may also become costly at large scale; reuse across Hamiltonians of the same active-space size can help mitigate that burden.Source: §5.3 · p. 62
  • Decomposed interactions [262] perform almost as well as GC with Sorted Insertion in measurements, with the performance gap narrowing as system size increases, and require lower depth: O(N) versus O(N^2).Ref. [262] reports that GC with Sorted Insertion tends to minimize measurements for a given precision; decomposed interactions use significantly less depth for the required unitary transformations.Source: §5.3 · p. 62
  • Basis rotation groups have more predictable implementation costs than FRO and its extensions, which require a minimization that may be computationally costly or reduce accuracy.Further research is needed to investigate these potential costs and accuracy losses.Source: §5.3 · p. 62
  • Inference methods may improve the listed costs, but methods requiring an additional machine-learning model can introduce training pitfalls, and thorough numerical comparisons remain a future research need.The review identifies decomposed interactions as having the most supporting arguments for treating molecular Hamiltonians.Source: §5.3 · p. 62

6. Ansatz selection and construction

This section reviews VQE ansatz design, including barren plateaus, initialization, and fixed, hardware-efficient, problem-tailored, and adaptive circuit constructions. It examines trade-offs among expressibility, accuracy, symmetry, circuit depth, and measurement and optimization costs, including unresolved scaling and convergence limits.

6. Ansatz selection and construction

  • An effective ansatz balances expressibility—the span of reachable states—with trainability, favoring a tractable subspace that contains states near the solution.High expressibility can make optimization intractable because of the parameters, iterations, or measurement shots required.Source: §6 · p. 63

6.1. The barren plateau problem

  • Barren plateaus make VQE optimization difficult: stochastic measurements struggle to resolve exponentially small gradients, potentially requiring exponentially many shots and also impeding gradient-free optimizers.The problem is characterized by gradients vanishing exponentially with the number of qubits under random initialization [86]; gradient-free methods can also fail when the cost landscape has too little variance [89] [444].Source: §6.1 · p. 64

6.1.1. Drivers of the barren plateau problem

  • Barren plateaus can arise when both ansatz segments are 2-designs, making gradient variance vanish exponentially with system size; high expressibility (low ε) lowers its bound [86] [123].Local encodings with lower Pauli weight offer more resilience than Jordan–Wigner encoding [87]. Noise-induced barren plateaus are independent of parameter initialization and cost-function locality, and involve vanishing expectation-value amplitude [442].Source: §6.1.1 · pp. 65–66

6.1.2. Methods to address barren plateau problem

  • Mitigation approaches include restricting ansatz expressibility, using adaptive ansätze or local encodings, and selecting initial parameters to avoid starting in a barren plateau [135].Grant et al. [135] propose a shallow random entangled initialization, but it can be challenging to implement and is not straightforward for some ansatz structures; other proposals include correlated parameters and layer-by-layer training.Source: §6.1.2 · pp. 66–67

6.1.3. Comments on barren plateau in the context of the VQE

  • For VQE, depth, parameter count, and entangling-gate count help compare ansätze, but only when they can accurately describe the ground state; no logarithmically scaling ansatz is known to guarantee that accuracy.Local encodings and logarithmic-scaling ansätze measured on local observables show some resilience, while local minima, increasing entanglement, or noise can undermine optimization [87].Source: §6.1.3 · p. 68

6.2. Fixed structure ansätze

  • Fixed-structure ansätze are set at the beginning of VQE and remain unchanged during optimization except for their parameter values.Qubit Coupled Cluster and Qubit Coupled Cluster Singles and Doubles are discussed with adaptive ansätze because their adaptive forms are more relevant there.Source: §6.2 · p. 68

6.2.1. Hardware-efficient ansatz (HEA)

  • The hardware-efficient ansatz repeats blocks of parameterized single-qubit rotations and entangling gates, offering device-specific flexibility and high expressibility [263].Its broad Hilbert-space coverage can be inefficient and may require exponential depth in the worst case.Source: §6.2.1 · p. 68
  • HEA is convenient for proof-of-principle studies but is unlikely to suit general large-scale chemical problems, partly because it needs many parameters and is vulnerable to barren plateaus.For exact ground-state resolution, the authors suspect the required depth L may grow exponentially with system size; HEA-type structures can nevertheless give reasonable spin-chain accuracy with at least linearly many layers.Source: §6.2.1 · pp. 69–70

6.2.2. The Unitary Coupled Cluster (UCC) Ansatz and extensions

  • UCC makes coupled-cluster evolution unitary through anti-Hermitian excitation operators, enabling variational quantum evaluation despite exponential classical evaluation cost.Its circuit depth is polynomial; a single Trotter step can suffice for accurate ground states of simple molecules, although operator ordering can affect accuracy.Source: §6.2.2 · pp. 71–72
  • UCCGSD removes occupied-to-virtual restrictions and has O(N^3) depth, with reported greater noise robustness and accuracy than UCCSD.Generalized UCC also improved DUCC results on H2O toward FCI, at the cost of additional parameters.Source: §6.2.2 · pp. 72, 74
  • k-UpCCGSD uses paired excitations and repeated ansatz blocks for O(kN) depth and O(kN^2) parameters, but increasing k complicates optimization and initialization affects results.Numerically, it finds better energies than UCCSD, with corroborating accuracy studies.Source: §6.2.2 · pp. 72–73
  • PNO-UpCCGSD restricts excitations to pair-natural orbitals and reduces parameter and CNOT counts by factors of 4 to 8 versus k-UpCCGSD.The PNO basis models relevant virtual states for orbital pairs and can substantially shrink the virtual orbital space.Source: §6.2.2 · p. 73
  • OO-UCC optimizes orbital rotations classically alongside quantum VQE, trading greater classical computation for simpler circuits and fewer qubits for correlations beyond the active space.Other variants include cluster-Jastrow circuits that avoid Trotterization and selective coupled cluster, which may require fewer measurements than ADAPT-VQE [513].Source: §6.2.2 · pp. 73–75
  • DUCC folds external excitations into an effective active-space Hamiltonian, and has recovered substantial correlation energy with quantum-resource savings.Using UCCGSD rather than UCCSD in this framework brought H2O results close to FCI, with additional parameters.Source: §6.2.2 · pp. 73–74
  • UCC-type ansätze use fewer parameters than HEA and can approach FCI accuracy, but are not hardware-efficient and may require large gate-count prefactors and connectivity overhead.UCCSD outperformed conventional CCSD in the reported systems, while orbital-optimized variants were within 10 to 20 mH of UCCSD with reduced quantum resources.Source: §6.2.2 · p. 75
  • UCC ansätze can require large circuit depth and entangling-gate counts, and their accuracy depends on fermionic-operator ordering, especially for highly correlated states near dissociation limits.Increasing k in k-UpCCGSD can largely address ordering sensitivity, while qubit-excitation and qubit-operator coupled-cluster variants may reduce CNOT scaling; both HEA and UCC can also produce PES kinks when symmetries are not preserved.Source: §6.2.2 · p. 75

6.2.3. Symmetry-preserving methods

  • Symmetry-preserving ansätze restrict the search space and may speed optimization, reduce barren-plateau risk, and prevent PES kinks associated with states of different electron number.Exchange-type gates preserve electron number under Jordan–Wigner and Parity mappings, but not generally under Bravyi–Kitaev; symmetry preservation can also be imposed through the cost function.Source: §6.2.3 · p. 76
  • The Efficient Symmetry-Preserving ansatz spans a broad symmetry-constrained Hilbert space, is less restrictive than UCC, and is described as closest to spanning FCI.Its CNOT cost can grow exponentially when the electron count is near half the number of orbitals, making it prohibitive in that regime despite possible advantages when electron count is negligible or close to the orbital count.Source: §6.2.3 · pp. 77–78

6.2.4. The Hamiltonian Variational Ansatz and extensions

  • HVA is well suited to many-body models, showing barren-plateau resilience and exponential convergence toward the Kagome-lattice ground state with circuit depth in noiseless simulations of up to 20 sites.Symmetry-breaking HVA variants report strong fidelities with reasonable entangling-gate counts up to 12 qubits and fewer gates than UCCSD for 12-qubit H2O, while VMFHA performs better where VEHA fails; Fourier-transform HVA has compact O(LN) scaling on planar lattices for jellium but has not been tested on alternative systems.Source: §6.2.4 · pp. 79–80

6.3. Adaptative structure ansätze

  • Adaptive-structure ansätze grow circuits by selecting operators according to their contribution to energy or another metric, and include iterative growth, Hamiltonian dressing, and variational gate-structure updates.Comparative metrics for these approaches remain difficult to establish reliably.Source: §6.3 · p. 80

6.3.1. Iterative ansatz growth methods (ADAPT-VQE and extentions)

  • For BeH2, ADAPT-VQE reaches 10^-3 kcal/mol error with about 50–60 parameters versus nearly 300 for UCCSD, but uses significantly more total measurements on average.It is also reported to resist barren plateaus, though pool gradients may vanish exponentially; gradient measurement cost and the need to reoptimize after each operator add to its expense.Source: §6.3.1 · p. 81
  • Nuclear ADAPT-VQE uses an O(N^2) operator pool and was tested up to N=12, showing reasonable outer-loop iteration scaling to reach good accuracy.The pool uses one- and two-body operators; tests covered the Lipkin–Meshkov–Glick model and nuclear shell model.Source: §6.3.1 · p. 82
  • Qubit-ADAPT uses Pauli-string operators to reduce gate costs, while QEB-ADAPT offers fewer CNOTs than both ADAPT variants with a parameter count similar to ADAPT-VQE.Qubit-ADAPT can remove Jordan–Wigner Pauli-Z chains and limit Pauli weight to four, but requires a larger operator pool and more parameters for a given accuracy; qubit-based ansätze may therefore be harder to optimize than fermion-based ones.Source: §6.3.1 · pp. 82–83, §6.4 · p. 88

6.3.2. Iterative Hamiltonian dressing (iterative Qubit Coupled Cluster (iQCC) and extensions)

  • iQCC fixes ansatz size by iteratively dressing the Hamiltonian, but convergence to the exact solution is not guaranteed and Hamiltonian terms can grow exponentially with iterations.Truncated Baker–Campbell–Hausdorff expansion limits per-iteration term growth to O(g^2); for active-space H2O, chemical accuracy required k=5–10 iterations depending on bond distance, and term growth can plateau in numerical analyses.Source: §6.3.2 · pp. 85–86
  • ClusterVQE separates within-cluster ansatz growth from cross-cluster Hamiltonian dressing, enabling shallower, separately optimized circuits and similar accuracy to iQCC with fewer Hamiltonian terms.Cluster circuits can run on smaller quantum computers and in parallel, but require measurements multiplied by the number of clusters; Hamiltonian growth and costly inter-cluster mutual-information clustering remain scaling concerns.Source: §6.3.2 · p. 86

6.3.3. Ansatz structure optimization

  • RotoSelect [129] chooses a rotation axis to improve convergence at added early sampling cost, while Variable Ansatz [534] stochastically adds and removes gates but has unclear implementation cost.RotoSelect’s reported results used HEA and may differ for UCC-based ansätze; Variable Ansatz was also shown to outperform HEA on Ising, Heisenberg, H2, and H4.Source: §6.3.3 · p. 87
  • Nearly twenty times shallower and with twelve times fewer CNOT gates than UCCSD, EVQE ansätze [548] were reported; MoG-VQE [549] reached chemical accuracy on 12-qubit LiH with 12 CNOTs.Machine-learning ansatz methods also target trainability and noise resilience, but their training costs and scaling for large systems remain largely untested.Source: §6.3.3 · p. 87

6.4. Discussion on ansätz used in the VQE context

  • k-UpCCGSD offers a strong accuracy–cost trade-off with linear scaling, while layered HEA may be unsuitable for large systems and particularly prone to barren plateaus.Problem-tailored or adaptively constructed ansätze, local Hamiltonian encodings, and suitable initialization are identified as design choices that may help VQE avoid barren plateaus, apart from noise-induced barren plateaus.Source: §6.4 · p. 88
  • Adaptive ansätze reduce parameter counts by less than an order of magnitude and show numerical resilience to barren plateaus, but require costly gradient estimates and repeated VQE optimizations.Gradient estimation scales with Hamiltonian size, while repeated optimizations are shallow early in the process but not near its end; their relative benefit needs further study.Source: §6.4 · p. 88
  • Computational-basis initialization has minor impact on HEA target-state fidelity versus Hartree–Fock, while UCC ansätze generally use Hartree–Fock initialization.A quantum routine for the Gutzwiller wavefunction [240] gave a better Fermi–Hubbard ground-state approximation than a mean-field method and may initialize lattice-model VQE; further preparation work appears in Ref.. The wavefunction is physically motivated for varied lattice systems.Source: §6.4 · p. 88

7. Optimization strategies

This section reviews VQE optimization strategies for finding sufficiently accurate ansätze, including gradient-based, second-order, gradient-free, landscape-based, and engineered-cost-function approaches. It describes their mechanisms, convergence and barren-plateau resilience, practical limitations, and trade-offs in measurement and implementation complexity.

7. Optimization strategies

  • VQE optimization is NP-hard and must achieve a good approximation in tractable steps despite noise, shot-limited precision, and potentially vanishing gradients.Sampling and gate noise can hinder convergence; the precision of measured expectations depends on shot count.Source: §7 · pp. 88–89

7.1. Background and notation

  • Quantum devices can yield expectation values by averaging single-shot bit-string measurements or, on some systems, directly averaging readout signals.The latter includes some NMR systems that read ensembles of molecules rather than individual states.Source: §7.1 · p. 90

7.2. Gradient evaluation

  • Many numerical optimizers require gradients, and parametrized-gate ansätze allow observable gradients to be evaluated directly on quantum computers.Gradient evaluation is important because many numerical optimization methods depend on objective-function derivatives.Source: §7.2 · p. 90, §7.2.2 · p. 91

7.2.1. Stochastic approximation methods

  • Stochastic approximation estimates expectation properties by random sampling; SPSA uses two objective measurements per iteration and can handle noisy optimization.Unlike FDSA’s single-direction perturbation, SPSA uses a random vector to perturb multiple variables.Source: §7.2.1 · p. 90

7.2.2. Analytical gradient calculation

  • Analytical gradient measurement uses direct or indirect observables; parameter-shift methods extend to higher derivatives, and applying the rule before fermionic encoding reduces VQE measurement requirements [562].The standard parameter-shift relation is limited to generators with exactly two eigenvalues; modified shifts or trigonometric interpolation address broader operators [124] [125].Source: §7.2.2 · pp. 91–93

7.3. Gradient-based searching strategy

  • Gradient-based strategies use first derivatives, while second-order optimizers also use second derivatives, at the cost of computing them.The section surveys popular gradient-based optimizers.Source: §7.3 · p. 93

7.3.1. First order optimizers

  • First-order methods include gradient descent, RMSProp’s adaptive scaling, and Adam’s momentum-based adaptive moments; Adam uses only first-order gradients and can handle noisy gradients [579].RMSProp divides gradients by a weighted moving average of their root-mean-square magnitude.Source: §7.3.1 · pp. 93–94

7.3.2. Second order optimizers

  • BFGS uses Hessian curvature to precondition gradients, but full Hessians are difficult to compute; quantum natural gradients instead descend using quantum information geometry [422–425, 580, 582–590].Pure-state quantum natural-gradient descent is equivalent to imaginary-time evolution [290].Source: §7.3.2 · pp. 94–95
  • Quantum natural gradient handles noisy, non-unitary circuits via a generalized Fubini–Study metric, but conditioning and Fisher-matrix construction are costly; SPSA approximation converges faster than standard descent but worse than the full method.Small Fisher-matrix eigenvalues can cause large updates; small initial learning rates, Tikhonov regularization, or half-inversion can stabilize them. Constructing the full metric for p parameters requires estimating p^2 expectation values.Source: §7.3.2 · pp. 96–97

7.4.1. Gradient-free optimizers

  • Nelder–Mead uses simplex-based gradient-free search with few measurements per iteration and performs well at low noise but less well with noise; Powell’s method searches conjugate directions using line minimization.Nelder–Mead iteratively replaces simplex vertices based on objective values; Powell’s line searches can use Golden-section search or Brent’s method.Source: §7.4.1 · pp. 97–98

7.4.2. Analytical optimization

  • Rotosolve exploits the sinusoidal dependence of linear objectives on individual parameters, fitting each from three measurements and iteratively setting it to its minimum.Its usual validity conditions exclude some ansätze; a generalized version using alternative parameter-shift rules supports arbitrary gate frequencies.Source: §7.4.2 · pp. 98–99
  • Fraxis fixes rotation angles and flexibly optimizes axes, outperforming Rotosolve in some numerical tests; a later method optimizes angle and axis together for time-evolution simulations.Fraxis is an alternative to methods that fix rotation axes and vary angles [272]; Wada et al. [273] extended the approach.Source: §7.4.2 · p. 99
  • Quantum analytical descent approximates the full linear-objective landscape using a quadratic number of parameters and measurements, then applies natural-gradient descent to find minima.The method iteratively updates parameters to a local minimum of the approximated landscape and repeats until convergence.Source: §7.4.2 · pp. 100–101
  • Jacobi-inspired optimization fits the landscape over small, iteratively selected parameter subsets, while DIIS uses prior iterations to improve estimates and convergence [563].The parameter subset is selected randomly; Anderson/Pulay approaches approximate the inaccessible optimization error using the cost-function gradient.Source: §7.4.2 · p. 101

7.5. Engineering cost function

  • Collective optimization jointly optimizes solutions across molecular bond distances as a parameter-space “snake,” and simulations show it can pull parameters from local minima.Its cost function accounts for the snake’s length and curvature.Source: §7.5 · pp. 101–102
  • CVaR emphasizes low-energy samples to smooth the objective without local minima [600], while symmetry penalties enforce constraints and can remove kinks without extra quantum measurement costs [38].Because CVaR discards samples, matching expectation-value accuracy requires more sampling. Symmetry constraints can complicate optimization and introduce risks of local minima.Source: §7.5 · pp. 102–103

7.6. Discussion

  • Numerical studies find that Rotosolve and its extensions converge significantly faster than other methods, despite requiring more measurement points per iteration than gradient descent strategies.Analytical methods are much faster per iteration, but that comparison does not account for their additional measurement points.Source: §7.6 · p. 104
  • On hydrogen, SPSA outperforms Nelder–Mead and Powell, and across small molecules it performs slightly better than four other optimizers [604].Source: §7.6 · p. 104
  • Covariance functions can accelerate near-converged optimization [605], while stochastic add-ons can reduce hardware-noise impact and speed convergence [606] [607] [608].Source: §7.6 · p. 104
  • Natural-gradient strategies are considered barren-plateau resilient, with an absolute gradient lower bound of 1/(2^(2N+1)); gradient-free analytical methods can avoid plateaus but cost more measurements and classical computation [126] [127] [128] [129] [130] [271].Source: §7.6 · p. 104
  • No optimizer excels in every aspect: convergence trades off against implementation complexity and measurement count, and multiple strategies can be combined.Sequential analytical fitting is mainly used with non-traditional ansatz optimizers, while quantum natural-gradient methods are more popular in simulation studies.Source: §7.6 · p. 104
  • Efficient circuit simulation can optimize local order parameters before quantum devices measure global quantities post-optimization [305].Source: §7.6 · p. 104

8. Error mitigation for VQE

This section reviews error-mitigation approaches for VQE, including symmetry verification, noise extrapolation, probabilistic error cancellation, exponential error suppression, and measurement-readout mitigation. It considers their costs and effects, including increased sampling variance and the influence of hardware noise on optimization and optimized results.

8. Error mitigation for VQE

  • At an optimistic error threshold of 10^-3, exploring interesting quantum chemistry problems with error correction could require about 10^6 physical qubits, alongside substantial ancilla, decoding, and communication overhead [233].Error correction could suppress noise arbitrarily when the noise is below threshold and enough qubits are available, but demonstrations have shown only limited fault-tolerant capabilities.Source: §8 · p. 104

8.1. Symmetry verification

  • Symmetry verification post-selects outcomes with the desired symmetry value, filtering symmetry-violating errors and potentially projecting the state into the symmetry-preserving subspace [274] [275].Its filtering benefit applies regardless of error rates, although failure rates rise with noise; experimental VQE demonstrations have shown measurement-noise filtering [156].Source: §8.1 · p. 105
  • Final symmetry verification is often available at no additional cost, bulk verification requires Pauli-product symmetries, and post-processing can avoid post-selection by measuring O, S, and OS [275].Final verification checks the symmetry after the ansatz; bulk verification checks during computation without disrupting it. Post-processing adds one measured observable OS per Pauli observable and one measurement of S overall.Source: §8.1 · pp. 105–106

8.2. Extrapolation based methods

  • Extrapolation estimates error-free expectation values by fitting measurements taken at multiple noise strengths, but requires extra circuit executions and can add noise when adjusting noise strength.For VQE, the review recommends fitting each Pauli observable separately because measurement circuits can have different noise, especially with grouping strategies; fitting per hardware can also help when noise varies across devices.Source: §8.2 · pp. 106–107

8.2.1. Method to systematically increase the noise

  • Noise can be amplified for extrapolation by directly rescaling the Hamiltonian, Pauli twirling Clifford entangling two-qubit gates, or replacing each CNOT with an odd number of consecutive CNOTs [623].With many CNOTs, higher-order errors quickly overshadow lower-order modeling terms. A Poisson-sampled variant, with (j_i−1)/2 drawn from a Poisson distribution of mean v, permits extrapolation to v=−1/2 and is more noise friendly when v is chosen so 1+2v is close to 1.Source: §8.2.1 · pp. 107–108

8.2.2. Modeling the noise and extrapolating from measured data

  • Polynomial extrapolation can overfit when its order is high, while exponential extrapolation can predict the exact outcome more accurately if sufficient sampling is affordable, despite typically higher variance.Extrapolation increases circuit repetitions and observable variance; the polynomial model approximates the exponential model when noise is not very high.Source: §8.2.2 · pp. 110–111

8.3. Probabilistic error cancellation

  • Probabilistic error cancellation modifies quantum circuits to invert noise effects and estimate ideal measurement values [276, 629].The estimate combines outcomes from different, slightly modified circuits with selected weights and operations.Source: §8.3 · p. 111
  • Probabilistic error cancellation samples modified circuits with quasi-probability weights, replacing an exponentially growing circuit set with a cost governed by estimator variance.For Pauli observables, the variance is bounded by γ_PEC^2 under the stated independence and bounded-outcome assumptions; tomography-based and learning-based constructions are described.Source: §8.3 · pp. 111–112

8.4. Exponential error suppression

  • Exponential error suppression amplifies the dominant eigenvector of a noisy VQE state, so energy minimization can yield an energy exponentially close to the true ground-state energy.The dominant state need not equal the ideal error-free circuit output; for some VQE-based algorithms, coherent mismatch matters, though its error is bounded by the limitation from creating more copies.Source: §8.4 · p. 114

8.5. Measurement readout error mitigation

  • Readout errors affect VQE accuracy, while encoding and measurement grouping shape their impact: low Pauli weight can reduce exposure, but grouped measurements may erase that advantage.Grouping can also increase the impact of an error; calibration experiments estimate a measurement matrix for correcting SPAM errors.Source: §8.5 · pp. 114–115

8.5.1. Measurement calibration matrix

  • The full measurement calibration matrix grows exponentially with measured qubits, whereas a no-crosstalk product approximation costs O(N) measurements and remains the best scalable approximation described.More elaborate schemes account for neighboring-qubit crosstalk but do not resolve the matrix's exponential growth.Source: §8.5.1 · p. 115

8.5.2. Correcting measurement outcomes

  • Corrected readout outcomes are obtained through a constrained optimization rather than direct matrix inversion, which can yield nonphysical results.The matrix and outcome vectors scale exponentially; a sparse, product approximation ignores correlated errors, and pathological calibration matrices can amplify uncertainty and cause oscillations.Source: §8.5.2 · pp. 115–116

8.5.3. Exploiting state-dependent noise

  • For state-dependent readout bias, splitting measurements into standard and qubit-inverted parts can average the bias, but may worsen performance for states favored by that bias.The adaptive alternative depends on predicting state probabilities and bias, limiting its generality; the review recommends evaluating the non-adaptive method on analytically tractable circuits first.Source: §8.5.3 · p. 116

8.6. Other error mitigation methods

  • Other approaches combine mitigation methods, detect errors, constrain reduced density matrices, model noise, or use machine learning and noise-aware circuit design.Error-detecting codes can restrict applicable gates, and the scalability of effective-noise models remains unclear; the review also describes enhanced sampling and second-derivative noise cancellation.Source: §8.6 · pp. 116–117

8.7. Impact of error mitigation on the scaling of VQE

  • Error mitigation raises VQE cost by increasing the variance of Pauli-term expectation estimates and the repetitions needed to meet a fixed statistical-error threshold.The review summarizes variance increases for specific methods in Table 12, but the supplied passage does not provide its values.Source: §8.7 · p. 117
  • Most error mitigation methods inevitably increase variance exponentially with qubit count or repeated ansatz layers because errors accumulate passively [96].The review notes that error mitigation differs from active error correction, which interacts with the quantum system to suppress errors.Source: §8.7 · p. 118
  • Reducing mitigation accuracy may lower variance, and tailoring mitigation to ground states or Hamiltonian eigenspaces may reduce overhead, but this adaptation remains unstudied [96] [661].The review also notes that mitigation costs depend on the noise, Hamiltonian, and sometimes the desired accuracy.Source: §8.7 · pp. 117–118
  • Applying error mitigation only as VQE optimization begins to converge could reduce its cost, but when to apply it requires further research.It remains unclear whether mitigation is needed during initial training.Source: §8.7 · p. 118
  • Under a shared noise model, symmetry verification had a better extraction rate than several compared methods, including probabilistic error cancellation and exponential error suppression [662].The fidelity boost metric differed across methods, symmetry verification's boost depends on available Hamiltonian symmetries, and the comparison omitted Richardson extrapolation methods.Source: §8.7 · p. 118

8.8. Noise robustness of VQE algorithms

  • VQE optimization can compensate for systematic coherent errors such as over-rotations, but optimized angles may be hardware-specific and need not transfer to another device [38] [663].A different angle can correct the implemented ansatz unitary, or correction may be unnecessary if it already produces the desired ground state.Source: §8.8 · p. 118
  • Hardware errors can reshape the cost landscape and create more local minima, while device-to-device noise variation challenges VQE parallelization [664].The review cites experimental investigation of this optimization issue in the related variational quantum factoring algorithm.Source: §8.8 · p. 118

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

This section extends VQE beyond ground states of isolated molecules, covering approaches to excited states and using VQE as a solver within multiscale methods that couple correlated subspaces to wider environments.

9.1. Excited states VQE

  • Quantum Subspace Expansion (QSE) diagonalizes a classically tractable, low-order-polynomial subspace built from a ground-state VQE, using Krylov [239] or low-rank excitation states.Reliable excited states require a subspace spanning the relevant low-energy excitations; noisy matrix elements can bias eigenvalues, while high-order reduced density matrices can be costly to sample [677].Source: §9.1 · p. 119
  • Direct variational methods modify the cost function to maintain orthogonality, avoiding subspace-expansion biases but typically requiring more quantum resources and a chosen ansatz; options include symmetry constraints and folded spectrum.Folded spectrum minimizes the energy of (H − γ1)^2 to target the eigenstate nearest γ, but may increase measurement demands and requires prior eigenspectrum knowledge; Liu et al. applied it to many-body localization.Source: §9.1 · p. 120
  • SSVQE trains one ansatz on orthogonal inputs to learn multiple states simultaneously, whereas VQD iteratively adds overlap penalties to compute successive excited states.SSVQE optimization may become increasingly constrained as more excited states are sought; VQD overlap evaluation can be costly or noisy, prompting lower-depth and adaptive variants [682].Source: §9.1 · p. 120
  • DVQE uses a generative adversarial network to enforce orthogonality before minimizing energy, while Variance VQE [686] combines variance and energy minimization to target low-lying excited states.DVQE discriminator depth scaling remains unknown; variance alone cannot select low-energy eigenstates because all eigenstates have zero variance, and an adaptive variant targets highly excited states [693].Source: §9.1 · pp. 120–121
  • VQE approaches can target dynamical correlation functions directly in time or frequency domains, describing response across the energy range but facing harder optimization and Hadamard-test requirements.The larger cost-function condition number complicates optimization, and transition amplitudes between response or excited states and the ground state require Hadamard tests.Source: §9.1 · p. 121

9.2. VQE as a solver of correlated subspaces in multiscale methods

  • CAS methods treat strongly correlated orbitals as an active space, with VQE solving its wavefunction while classical orbital rotations can optimize the space; CASPT2 coupling to external orbitals may require costly higher-order density matrices.CAS-based approaches target strongly correlated systems including competing spin states and bond-breaking geometries; optimizing the active space requires its two-body reduced density matrix, while CASPT2 may require three-body or higher matrices [709].Source: §9.2 · pp. 121–122
  • DMET embeds a local fragment with a minimal bath and matches one-body density matrices, while DMFT includes local fluctuations through a self-consistent self-energy and can use VQE-type impurity solvers.VQE-DMET has been implemented for applications including protein-ligand interactions, and energy-weighted DMET captured quantum phase transitions beyond DMET; DMFT solvers sample Green’s functions in time or frequency domains [695] [697].Source: §9.2 · pp. 122–123
  • Embedding methods use VQE to solve a correlated subspace self-consistently within a wider environment, recovering electron correlation while reducing qubit requirements at the cost of classical resources and a self-consistent loop.The review identifies embedding and broader multi-method hybrid schemes as promising ways to extend quantum computation toward practical molecular modeling.Source: §9.2 · p. 123

10. Conclusion and outlook

The review finds VQE promising for near-term quantum computing, but its practical applicability depends on choices across the algorithm and on resolving substantial open questions. It surveys these components, related algorithms, and priorities for future research.

Hamiltonians and mappings

  • Low-weight fermion-to-spin mappings can improve circuit depth and barren-plateau resilience, while Hamiltonian and basis choices affect qubit requirements and accuracy.Canonical orbital bases remain dominant, with plane-wave and frozen-natural-orbital alternatives; ternary-tree mapping is highlighted for molecular Hamiltonians and generalized superfast encoding for lattice models.Source: §10 · p. 123

Measurement

  • Joint measurement methods are needed to substantially reduce VQE measurement scaling: decomposed interactions suit molecular Hamiltonians, while qubit-wise commutativity grouping may suit lattice models.Shot allocation weighted by Hamiltonian coefficients and truncating terms can also save measurements.Source: §10 · p. 123

Ansätze

  • The k-UpCCGSD ansatz offers high accuracy with linear scaling in qubit number, while Hamiltonian Variational Ansatz methods suit lattice models and adaptive methods have uncertain cost and scaling.The k-UpCCGSD extension is also noted; the Hamiltonian Variational Ansatz and extensions are promising for many-body physics.Source: §10 · p. 123

Optimization and noise

  • Evidence for optimizer performance remains sparse, with quantum natural gradient and RotoSolve having numerical support; noise mitigation may require costly symmetry verification or extrapolation.Variational algorithms can learn some biases caused by quantum noise, but the review says this may not suffice for target accuracy.Source: §10 · p. 123

Future research

  • Comprehensive comparative applied studies remain rare, and the review prioritizes research on measurement shots, parallelization, barren plateaus and optimization, and tractable noise mitigation.Further work should assess barren-plateau mitigation, optimization landscape complexity, convergence rates, and optimizer roles in avoiding local minima.Source: §10 · p. 124

Related algorithms

  • VQE-inspired methods alter optimization, reduce qubit or communication costs, target other eigenproblems, or adapt circuits, including FQE, QAS [726], and ctrl-VQE [737].Projective Quantum Eigensolver uses Schrödinger-equation residuals; Quantum Sampling Regression [729] samples the energy landscape’s Fourier basis; Deep VQE [734] divides a Hamiltonian into sub-units; VQSE [730] targets density-matrix eigenpairs. Filtering VQE reports outperforming standard VQE and QAOA for a quadratic unconstrained binary optimization Hamiltonian.Source: §10 · p. 124

Outlook

  • VQE may be paired with other algorithms, including QPE, but practical calculations require overcoming daunting obstacles despite potential applications in drug discovery, chemical engineering, and materials science.The review describes VQE as among the most promising methods for near-term quantum computing applications.Source: §10 · p. 124

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

The sections describe how fermionic qubit encodings can be represented and generalized with Fenwick trees, then explain how the Hadamard test measures the real or imaginary part of ⟨ψ|U|ψ⟩.

A. Qubit encodings and Fenwick trees

  • Bravyi–Kitaev Fenwick trees are built recursively by joining two copies of the smaller tree, mirroring the change-of-basis construction.The single-qubit encoding is the trivial graph; each larger power-of-two tree joins the root-side vertex of one copy to the corresponding vertex in the other.Source: A. Qubit encodings and Fenwick trees · p. 125
  • In the Bravyi–Kitaev tree, a qubit’s update set comprises its ancestors, its flip set its children, and its remainder set qualifying children of its ancestors.For the remainder set, the child’s value must be less than the qubit’s value.Source: A. Qubit encodings and Fenwick trees · p. 125
  • The tree-based qubit sets yield creation and annihilation operator representations, and for n not a power of 2, sets are built at the next power of 2 before qubits ≥ n are discarded.The operator representations are recovered using Eq. (58).Source: A. Qubit encodings and Fenwick trees · pp. 125–126
  • Parity encoding corresponds to a linear graph, while Jordan–Wigner corresponds to a disconnected graph; collections of Fenwick trees can also define new fermionic encodings.For disconnected trees, the remainder set includes qualifying roots as well as children of ancestors; the parity and Jordan–Wigner operators are recovered by inspection with Eq. (58).Source: A. Qubit encodings and Fenwick trees · p. 126, Figure 20 · p. 126

B. Hadamard test

  • The Hadamard test uses an ancilla, Hadamard gates, and controlled-U to estimate the amplitude ⟨ψ|U|ψ⟩, including its real and imaginary parts.The real-part circuit begins with |ψ⟩⊗|0⟩, applies a Hadamard to the ancilla, then controlled-U and a second Hadamard before measurement.Source: B. Hadamard test · p. 126
  • For the real-part circuit, the ancilla’s probability difference P(0) − P(1) equals Re⟨ψ|U|ψ⟩; the second circuit’s difference equals Im⟨ψ|U|ψ⟩ [76].In the real-part circuit, P(0) = (1 + Re⟨ψ|U|ψ⟩)/2 and P(1) = (1 − Re⟨ψ|U|ψ⟩)/2.Source: B. Hadamard test · p. 126

C. Error mitigation appendix … C.1.4. Dephasing noise model

The appendix surveys common quantum-noise models, outlining relaxation, gate over-rotation, depolarization, and dephasing. It notes that dephasing-related noise may evade symmetry verification.

C.1. Common noise models

  • The appendix directs readers to Ref. for further details on common quantum-computing noise concepts.Source: §C.1 · p. 126

C.1.1. Relaxation rates

  • T1 and T2 characterize single-qubit state errors from thermalization and serve as widely available experimental metrics for quantum computers.T1 and T2 are also called longitudinal and transverse relaxation rates, respectively.Source: §C.1.1 · p. 127

C.1.3. Depolarizing noise model

  • The depolarizing model changes an n-qubit state ρ, with ε characterizing the noise strength.Source: §C.1.3 · p. 127

C.1.4. Dephasing noise model

  • Dephasing is associated with the disappearance of density-matrix off-diagonal elements and is closely related to a quantum computer’s T2 relaxation rate.Source: §C.1.4 · p. 127
  • The passage says depolarizing noise leaves a single-qubit state’s diagonal part unaffected, commutes with most symmetry operators, and cannot be detected by symmetry verification.Source: §C.1.4 · p. 127

C.1.5. Damping error

Damping error reduces the |1⟩ population of a single-qubit state and is closely related to T1 relaxation. Its strength is characterized by ε.

C.1.5. Damping error

  • Damping error reduces the |1⟩ population of a single-qubit state, is related to T1 relaxation, and has strength parameter ε.The model is described by Kraus operators; the passage does not provide their form.Source: §C.1.5 · p. 128

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

The sections illustrate probabilistic error cancellation (PEC) for single-qubit depolarizing noise and introduce a quantum-computer approach to computing the value in Eq. (304), focusing on its numerator.

C.2. Example for probabilistic error cancellation

  • The depolarizing channel is described using its noise strength and Pauli matrices [276].Source: §C.2 · p. 128
  • For an effective channel N1∘U, the example constructs four circuits: U and X∘U, Y∘U, and Z∘U.Source: §C.2 · p. 128
  • The measurement outcomes from these circuits are combined according to the weights in Eq. (323).Source: §C.2 · p. 128
  • The weighted construction represents U acting on the input state through quasi-probabilities over circuits appended with Pauli gates.Source: §C.2 · p. 128
  • PEC cancels the effect of depolarizing noise on a one-qubit quantum computer.The passage presents this as an illustrative example.Source: §C.2 · p. 128

C.3. Implementation of exponential error suppression

  • The implementation discussion considers quantum-computer methods for computing the numerator of the value in Eq. (304).Source: §C.3 · p. 129
  • The denominator is treated as a special case of the numerator with the identity observable.Source: §C.3 · p. 129

C.3.1. Ancilla assisted method

The ancilla-assisted method estimates Tr[ρ^K O] by expressing an observable on one system as an overlap between the original state and a permuted state across copies.

C.3.1. Ancilla assisted method

  • With an extra ancilla and K copies of ρ, the method measures O on one system through an overlap that yields Tr[ρ^K O].The overlap is between the original state and its permuted version.Source: §C.3.1 · p. 129
  • The permutation s is a K-cycle derangement of distinct indices, whose permutation constraints enforce the delta functions in the overlap expression.For a statistical mixture of K copies, this construction gives Tr[ρ^K O].Source: §C.3.1 · p. 129
  • The derangement unitary D_K implements the permutation; the overlap can then be measured with a modified Hadamard test.For the example cycle, D_K uses CNOTs on pairs (1, 2), …, (K−1, K), and (K, 1), as illustrated in Fig. 17a.Source: §C.3.1 · p. 129
  • Derangement improvements include a deep-circuit method on a single copy [747]; although long-range D_K may be error-prone, errors preserving orthogonality can be mitigated by extrapolation.The single-copy method avoids creating K copies and applying derangement across them; other improvements are discussed in [633] [747].Source: §C.3.1 · p. 129

C.3.2. Diagonalization method

The diagonalization method reformulates derangement as a swap matrix and estimates the numerator and denominator using unitary diagonalizations. It avoids an additional ancilla and long-range controlled operations, relying instead on local operations across copies of the state.

C.3.2. Diagonalization method

  • The method implements a unitary version of the derangement swap and the observable after derangement using local operations, without an additional ancilla or long-range controlled operations.The local operations connect the same sets of qubits in each copy of the state.Source: §C.3.2 · p. 130
  • Unitary diagonalizations of the swap matrix and observable-after-swap operator enable computational-basis measurements of the copies to estimate the numerator and denominator.Although these operators are not generally Hermitian, the passage notes that in many cases they can be diagonalized by unitary matrices.Source: §C.3.2 · p. 130
  • Because the swap factorizes over corresponding qubits, diagonalization simplifies for qubits outside the observable’s support; for a one-qubit observable, a commuting symmetrized observable yields the same factorized diagonalizer.The shared diagonalization unitaries act locally on corresponding qubits across the copies.Source: §C.3.2 · p. 130
  • Grouping methods can reduce complex multiple-qubit measurements to simple one-qubit measurements, making the implementation more appealing while retaining local qubit connections.The passage points to grouping methods discussed in Sec. 5.Source: §C.3.2 · p. 130

C.3.3. Dual state purification method

For K=2, dual state purification offers exponential error suppression without multiple state copies or controlled-unitary gates, at the cost of doubled circuit depth and final measurement of all qubits. Its circuit combines post-selected measurement probabilities to approximate a purified-state observable.

C.3.3. Dual state purification method

  • For K=2, dual state purification achieves exponential error suppression without multiple state copies or controlled-unitary gates, but doubles circuit depth and requires final measurement of all qubits [635].For general even K, the approach can be extended, but exponential error suppression without controlled-unitary overhead is achieved only for K=2 [636].Source: §C.3.3 · p. 131
  • The circuit applies U then U† and combines post-selected all-zero outcomes with mid-measurement-conditioned probabilities to approximate Tr(Z1ρ²)/Tr(ρ²).A separate circuit omits the mid-circuit measurement; under noise, applying U and U† there does not have an identity effect.Source: §C.3.3 · p. 131, Figure 23 · p. 131
  • The method extends to any observable by expanding it as a linear combination of Pauli observables, with an explicit circuit extension for observables satisfying O² = I [635, 636].Source: §C.3.3 · p. 131

C.3.4. Shadow tomography based method

The method uses classical shadows to estimate the numerator Tr(ρK̂O), avoiding extra copies or doubled circuit depth, but its sampling cost may grow exponentially with qubit count.

C.3.4. Shadow tomography based method

  • Refs. [748] [749] estimate Tr(ρK̂O) using classical shadows formed by applying random unitaries to ρ and measuring in the computational basis.Ref. [749] adds error-correction codes; Ref. [748] gives an explicit formula when the random-unitary pool satisfies the 3-design property.Source: §C.3.4 · p. 132
  • Unlike the preceding methods, shadow tomography requires neither additional copies of the same state nor a doubling of circuit depth.The passage contrasts these requirements with the methods in Secs. C.3.1–C.3.3.Source: §C.3.4 · p. 132
  • The required random-unitary samples may scale exponentially with qubit count; a preliminary study for K = 2 found lower cost than state tomography but impracticality at large scale.The passage gives the preliminary estimate as N_U≈20.82N.Source: §C.3.4 · p. 132

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