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Hybrid quantum-classical algorithms and quantum error mitigation

Suguru Endo, Zhenyu Cai, Simon C. Benjamin, Xiao Yuan

arXiv:2011.01382v1quant-ph

TL;DR

NISQ devices are noisy and limited, so determining useful tasks and reliable computational strategies remains important. This review synthesizes hybrid quantum-classical variational algorithms and quantum error mitigation, covering optimisation, simulation, and mitigation approaches. It concludes that shallow variational circuits are tailored to NISQ devices and support applications across optimisation, dynamics, and error suppression.

  • Problem

    Current NISQ devices have limited qubits and noisy gates, while useful tasks and reliable computational strategies for this regime remain to be established.

  • Method

    The paper reviews variational quantum optimisation, variational quantum simulation, their extensions, and quantum error mitigation techniques.

  • Results

    The review presents shallow variational algorithms as tailored to NISQ devices, with applications including Hamiltonian spectra, machine learning, linear algebra, open-system simulation, and Gibbs-state preparation.

  • Takeaways & Limitations

    Hybrid variational algorithms and error mitigation provide a reviewed framework for using noisy intermediate-scale quantum hardware before universal fault-tolerant quantum computers exist.

Abstract

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Quantum computers can exploit a Hilbert space whose dimension increases exponentially with the number of qubits. In experiment, quantum supremacy has recently been achieved by the Google team by using a noisy intermediate-scale quantum (NISQ) device with over 50 qubits. However, the question of what can be implemented on NISQ devices is still not fully explored, and discovering useful tasks for such devices is a topic of considerable interest. Hybrid quantum-classical algorithms are regarded as well-suited for execution on NISQ devices by combining quantum computers with classical computers, and are expected to be the first useful applications for quantum computing. Meanwhile, mitigation of errors on quantum processors is also crucial to obtain reliable results. In this article, we review the basic results for hybrid quantum-classical algorithms and quantum error mitigation techniques. Since quantum computing with NISQ devices is an actively developing field, we expect this review to be a useful basis for future studies.

I. INTRODUCTION

NISQ hardware is constrained by shallow circuits, limited qubits, and non-negligible gate errors, motivating hybrid quantum-classical algorithms and quantum error mitigation. This review introduces variational optimisation and simulation methods, their applications, and techniques for reducing computation errors.

  • Motivation: 53 qubits demonstrated quantum supremacy, but practical quantum advantage remains an unresolved goal.
  • Motivation: NISQ devices control tens to thousands of noisy qubits, with gate errors potentially on the order of 10^-3 or lower.
  • Motivation: Hybrid quantum-classical algorithms combine quantum processors with classical computation, reducing the need for fully coherent deep circuits.
  • Motivation: Quantum error mitigation post-processes experimental data without encoding qubits as full error correction, saving qubits for NISQ simulation.
  • Scope: The review covers variational quantum eigensolvers, variational quantum simulation, extensions to optimisation and physics problems, and error mitigation methods.
  • Variational algorithms: Variational quantum optimisation minimises problem-specific cost functions, whereas variational quantum simulation models quantum dynamics and related many-body processes.

III. VARIATIONAL QUANTUM OPTIMISATION

Variational quantum optimisation maps problems to Hamiltonians or cost functions whose ground states encode solutions, then uses parameter optimisation to approach those states. QAOA applies this strategy to classical optimisation by interpolating between an initial and problem Hamiltonian.

  • Variational optimisation constructs a Hamiltonian or cost function so the desired solution corresponds to a ground state or cost minimum.
  • QAOA: QAOA maps a classical optimisation problem to a Hamiltonian whose ground state represents the solution.
  • QAOA: Boolean satisfiability can be encoded clause by clause into a Hamiltonian whose ground state satisfies all clauses.
  • QAOA: QAOA uses an ansatz with alternating parameterised operations and can gradually change the Hamiltonian during optimisation.
  • QAOA: The interpolation starts from H(0) = H_X and ends at H(T) = H_P, emulating adiabatic state preparation as D increases.
  • QAOA: QAOA has been studied on MaxCut and implemented with 40 trapped-ion qubits.

B. Variational algorithms for machine learning

Variational quantum circuits extend machine-learning models to supervised, generative, and adversarial settings. They may improve representation of multipartite correlations and support sampling, but comparative advantage over classical neural networks remains unresolved.

  • Quantum neural networks use variational circuits that may represent multipartite correlations unavailable to efficient classical models.
  • Unitary quantum circuits preserve state norm, potentially providing natural parameter regularisation against overfitting.
  • Quantum circuit learning: Quantum circuit learning performs supervised learning by preparing an input-dependent variational state and measuring it to generate outputs.
  • Quantum circuit learning: QCL can introduce nonlinear models through state encoding, measurement, and classical post-processing.
  • QCL’s ability to outperform classical neural networks still requires further study.
  • Data-driven quantum circuit learning: DDQCL learns a data distribution with a variational quantum circuit, producing probabilities through Born-rule measurement.
  • Data-driven quantum circuit learning: A 4-qubit trapped-ion experiment successfully learned GHZ states and coherent thermal states using DDQCL.
  • Quantum generative adversarial networks: Quantum GANs replace classical neural-network generator and discriminator components with quantum neural networks.

4. Quantum autoencoder for quantum data compression

Quantum autoencoders compress quantum-state ensembles by encoding information into fewer qubits and decoding it back, while variational eigensolvers diagonalise density matrices and support related linear-algebra tasks.

  • Quantum autoencoder: Successful compression and decoding require recovery of the input within a desired accuracy ε.
  • Quantum autoencoder: Quantum autoencoders compress input states from n + m qubits to n qubits using a parameterised circuit and post-selection on m qubits.
  • Quantum autoencoder: Compression quality is assessed by a cost function computed with a SWAP test or destructive SWAP test.
  • Quantum autoencoder: The encoder compresses data and the decoder reconstructs it to the original dimension.
  • Variational quantum state eigensolver: VQSE diagonalises input density matrices by variationally rotating them and measuring computational-basis probabilities.
  • Variational quantum state eigensolver: At equality, measurement probabilities match the eigenvalues, achieving diagonalisation; measured basis states identify eigenvectors through the optimised circuit.
  • Linear algebra: Variational algorithms also formulate matrix-vector multiplication and linear-system solving as ground-state problems.
  • Linear algebra: Matrix-vector multiplication can be extended to Hamiltonian simulation by repeatedly applying approximations to short-time evolution operators.

D. Excited state-search variational algorithms

Variational excited-state methods transform or expand the problem so excited states can be obtained through ground-state optimisation or subspace eigenvalue problems. Their accuracy depends on the quality of the initial ground state, ansatz, and expanded subspace.

  • Motivation: Excited-state spectra support studies of many-body physics, chemical reaction dynamics, photodissociation rates, and absorption bands.
  • Overlap-based method: The overlap-based method penalises previously found eigenstates so successive excited states become ground states of modified Hamiltonians.
  • Overlap-based method: Overlap penalties can be measured with SWAP tests requiring two state copies, or through a reference-state return probability.
  • Overlap-based method: If the discovered ground state is inaccurate, all subsequently calculated excited states may be incorrect.
  • Quantum subspace expansion: Quantum subspace expansion approximates excited states as linear combinations in an expanded subspace and solves a generalised eigenvalue problem.
  • Quantum subspace expansion: Subspace-expansion spectra are error-mitigated because the excited states are represented within the expanded subspace.
  • Contraction VQE methods: SSVQE and MC-VQE first project onto a low-energy subspace, then respectively optimise state assignments or solve for expansion coefficients.
  • Contraction VQE methods: Contraction VQE methods may have more complicated energy landscapes, but averaging optimisation over multiple states can equalise excited-state accuracy.

4. Calculation of Green’s function

The review describes variational approaches for calculating Green’s functions through excited-state transition amplitudes, alongside circuit-recompilation and variational metrology applications.

  • Green’s-function calculation: Green’s functions can be obtained by calculating transition amplitudes between a ground state and excited eigenstates.The Lehmann representation connects these amplitudes to the Green’s function.
  • Green’s-function calculation: The review discusses contraction VQE, overlap methods, and MC-VQE for estimating transition amplitudes.The overlap method was generalized from a specific Jordan–Wigner encoding to general operators.
  • Circuit recompilation: Circuit recompilation variationally tunes hardware-compatible circuits to approximate target unitaries or their action on specific input states.Average gate infidelity and Hilbert–Schmidt-based costs provide recompilation objectives.
  • Variational-state quantum metrology: Variational-state quantum metrology found a highly asymmetric 9-qubit probe state that outperformed previous results.The optimised state can be obtained without knowing the quantum device’s noise model, according to the review.

G. Variational quantum algorithms for quantum error correction

Variational methods tailor quantum error-correction circuits and codes to hardware constraints or noise. The reviewed approaches include variational circuit compilation for target code states and QVECTOR for learning noise-preserving codes.

  • Motivation: Hardware-friendly quantum error correction must account for restricted operations and qubit topology on NISQ hardware.Conventional QEC formulations generally do not incorporate these experimental implementation constraints.
  • Variational circuit compiler: A variational circuit compiler discovers hardware-compatible circuits satisfying user-specified requirements for a given QEC code.Requirements can include available gate sets, limited topology, and achievable error rates.
  • Variational circuit compiler: The compiler designs a Hamiltonian whose ground state is the target logical state, then prepares it using VQE or variational imaginary-time simulation.Stabilizer generators can define the code space and contribute to the Hamiltonian construction.
  • Variational circuit compiler: If the discovered energy is sufficiently close to the ground-state energy, the resulting encoding circuit approximates the target QEC code well.The method was numerically tested on five- and seven-qubit codes with different gate sets in noiseless and noisy circuits.
  • QVECTOR: QVECTOR learns device-tailored encoding and decoding circuits that preserve quantum states under noise.It uses noisy encoding, recovery, and decoding operations and evaluates average fidelity over Haar-distributed input states.
  • QVECTOR: Under phase damping, QVECTOR learned a three-qubit code with a six-times-longer T2 than conventional methods.Under combined amplitude and phase damping, it learned a code outperforming the five-qubit stabilizer code.

I. Other applications

The review extends variational simulation beyond standard time evolution to mixed states, open-system dynamics, linear algebra, and stochastic Schrödinger trajectories. These applications include numerical demonstrations and resource reductions for open-system simulation.

  • Mixed-state simulation: Variational simulation generalizes from pure to mixed states by parametrizing density matrices and evolving their parameters using McLachlan’s principle.The review applies this framework to real and imaginary time evolution of mixed states.
  • Open-system simulation: Open-system real-time evolution can be simulated by parametrizing density matrices under the Lindblad master equation.The variational equations follow from minimizing the deviation between parametrized-state evolution and Lindblad dynamics.
  • Open-system simulation: Purification-based evaluation of open-system dynamics requires 4Nq qubits because two copies of a 2Nq-qubit purification are used.The required terms can be evaluated with SWAP-test circuits.
  • Generalised time evolution: Generalised time evolution unifies real and imaginary time evolution and describes first-order differential equations with non-Hermitian Hamiltonians.The review applies this framework to linear algebra and stochastic Schrödinger equations.
  • Linear algebra: Efficiently simulating generalised time evolution enables matrix multiplication and solving linear systems of equations.The constructions evolve states toward M|u0⟩ and M^-1|u0⟩, respectively.
  • Stochastic open-system dynamics: A stochastic Schrödinger-equation approach simulates open-system trajectories as mixtures of pure-state evolutions and jumps.Local Lindblad operators can be decomposed into unitary and diagonal components for variational simulation.
  • Stochastic open-system dynamics: A 6-qubit 2D Ising simulation witnessed a dissipation-induced phase transition using the stochastic Schrödinger-equation approach.The algorithm uses a single state copy and therefore requires one quarter as many qubits as the preceding method.

C. Gibbs state preparation

Variational imaginary-time simulation prepares Gibbs states by purifying a maximally mixed state and evolving a joint system–ancilla ansatz. The review also connects variational simulation to finite-temperature Green’s-function calculations and surveys error-mitigation principles relevant to noisy devices.

  • Gibbs-state construction: A maximally mixed state I_d/d can be purified as |ψ_max⟩ = 1/√d Σ_i |i⟩_s|i⟩_a, whose system marginal is I_s/d.The target Gibbs state is obtained by imaginary-time evolution of the system component.
  • Gibbs-state construction: A joint unitary ansatz variationally simulates imaginary-time evolution on the purified state to prepare the Gibbs state.The ansatz circuit acts on both system and ancilla registers, as illustrated for two register qubits.
  • Applications: The prepared Gibbs state can be combined with variational real-time simulation to compute finite-temperature Green’s functions.The ground-state workflow first approximates the state and then evaluates time-evolved operator correlations; Gibbs preparation extends this to finite temperature.
  • Error mitigation context: Quantum error mitigation targets ideal measurement statistics through classical post-processing rather than recovering the entire ideal output state.Such processing generally amplifies variance, requiring more samples, and is typically effective only when the whole-circuit error rate is sufficiently low.
  • Error mitigation context: Error extrapolation estimates the error-free result from measurements at several boosted error rates using a fitting model such as Richardson extrapolation.Increasing the extrapolation order suppresses calculation error from ε to O(ε^(n+1)), but the associated shot-noise cost increases.

2. Exponential extrapolation

Exponential extrapolation models noisy expectation values using the exponential decay associated with accumulated stochastic errors, rather than a polynomial error expansion. The review describes noise-amplification methods and notes both practical advantages and assumptions limiting their reliability.

  • Limitations: Polynomial Richardson and least-squares extrapolation assume a valid Taylor expansion with negligible higher-order terms, which may fail for small ε and large N_g.This motivates exponential expansions for practical circuits where polynomial approximations can become inaccurate.
  • Exponential model: For many gates with small Markovian stochastic error ε, the accumulated error expansion contains an exponentially decaying factor e^(-N_gε).The binomial distribution of errors is approximated by a Poisson distribution when N_g is large and N_gε = O(1).
  • Exponential model: Exponential extrapolation estimates ⟨M⟩(0) from measurements at ε and αε by fitting the first-order exponential expansion.The method uses two noise rates with α > 1 and defines an associated mitigation cost through the amplified sampling variance.
  • Reported applications: Exponential extrapolation was numerically advantageous over linear Richardson extrapolation and was demonstrated on IBM superconducting qubits for dynamical mean-field-theory simulation.It was later generalized to multi-exponential extrapolation for cases not fitted by a single exponential curve.
  • Noise amplification: Error rates can be boosted by inserting repeated noisy gates, rescaling Hamiltonian pulses, or applying Pauli twirling.For repeated CNOTs, 2n + 1 operations approximately boost the physical error from ε to (2n + 1)ε under commuting-noise assumptions.
  • Noise amplification: Noise amplification is model-dependent: noncommuting noise, non-rescaling-invariant dissipation, high single-qubit error, or poorly known channels can distort the intended boosted rate.These deviations can introduce estimation errors or limit the applicability of the amplification procedures.

4. Mitigation of algorithmic errors

Algorithmic errors from finite-step Trotterization can be treated as extrapolation parameters alongside physical noise. The review also describes quasi-probability mitigation, which cancels noise through probabilistic inverse-channel decompositions but has stringent cost and characterization requirements.

  • Algorithmic-error mitigation: Finite Trotterization introduces algorithmic error ε_A = 1/N_T, while increasing N_T reduces algorithmic error but adds physical noise.Choosing N_T therefore requires balancing algorithmic and physical errors.
  • Algorithmic-error mitigation: Algorithmic errors can be extrapolated by evaluating observables at boosted ε_A values produced with fewer Trotter steps.In practice, physical errors are mitigated first, followed by extrapolation of the algorithmic error.
  • Quasi-probability method: Quasi-probability mitigation represents an inverse noise channel as a signed combination of implementable basis operations and averages their measurement outcomes.Each operation is sampled according to probabilities proportional to the absolute decomposition coefficients, and outcomes are multiplied by coefficient signs.
  • Quasi-probability method: For stochastic noise, quasi-probability mitigation has cost γ_tot ≈ e^(2bεN_g), so it is efficient only when εN_g = O(1).The cost grows exponentially with the average number of errors in the entire circuit.
  • Quasi-probability method: Unlike extrapolation, quasi-probability mitigation requires exact identification of the circuit noise model.State-preparation and measurement errors can corrupt process estimates, although gate-set tomography can address such errors in the decomposition.

D. Quantum subspace expansion

Quantum subspace expansion mitigates variational-state errors by optimizing within a subspace generated from measured operators, with the optimization reduced to a classical generalized eigenvalue problem. Symmetry verification provides a related projection-based strategy, while post-processing trades additional samples for avoiding ancilla-based checks.

  • Quantum subspace expansion: Quantum subspace expansion minimizes the energy over states generated by applying a small operator set to an approximate VQE or imaginary-time state.The resulting optimization is reformulated as a generalized eigenvalue problem.
  • Quantum subspace expansion: The matrices required for the generalized eigenvalue problem are measured on a quantum computer and solved classically when the subspace is small.The optimized coefficients can then be used to evaluate expectation values of other operators.
  • Quantum subspace expansion: QSE is more effective for coherent errors such as gate over-rotation and may not mitigate all local stochastic errors.Its state-rotation component addresses coherent errors, while projection can remove stochastic components when the target state has a suitable symmetry or structure.
  • Symmetry verification: Symmetry verification discards noisy outcomes that leave the symmetry-preserving subspace, using parity-check circuits or equivalent post-processing.Particle-number and spin symmetries can be mapped to measurable Pauli operators.
  • Symmetry verification: Post-processing symmetry verification avoids additional ancillas and parity-check circuits but requires more samples.The method reconstructs measurements on the correctly symmetric subspace from expectation values of the Hamiltonian and symmetry operator.
  • Individual error reduction: Individual error reduction suppresses errors to first order using QEC on one qubit combined with post-processing, but is harder to implement than other QEM methods.Higher-order effects can be mitigated by combining it with additional error-mitigation methods.

G. Measurement error mitigation

Measurement error mitigation models how noisy measurement probabilities differ from ideal probabilities and corrects this transformation, while learning-based methods infer ideal results from efficiently generated Clifford training data. These approaches can avoid exact noise tomography in some settings but may require truncation or optimisation when recovery spaces grow exponentially.

  • Measurement error mitigation: Measurement errors transform the ideal probability distribution through a noise-dependent matrix estimated by detector tomography.Without detector crosstalk, the estimated matrix can factor into a tensor product of single-qubit transformations.
  • Measurement error mitigation: Constrained estimation enforces normalized, nonnegative probabilities when direct matrix inversion produces unphysical negative components.The issue can arise from non-classical noise such as coherent errors.
  • Learning-based mitigation: Clifford Data Regression learns x_ideal = g(x_noisy, θ) from classically simulated ideal and experimentally sampled noisy Clifford results, then predicts ideal outputs for general circuits.The regression can use a linear ansatz or a neural network and was demonstrated with a 16-qubit IBMQ computer and a 64-qubit classical simulator.
  • Learning-based mitigation: Learning-based quasi-probability mitigation fits recovery-operation coefficients using Clifford training circuits, avoiding direct process tomography of the error channel.The procedure can also apply to arbitrary non-Clifford single-qubit gates with gate-independent errors.
  • Learning-based mitigation: The recovery-operation space may grow exponentially with circuit size, motivating Pauli-twirling truncation or Monte Carlo variational optimisation.Truncation reduces the space to a subset whose dimension grows polynomially with circuit size.

I. Stochastic error mitigation

Stochastic error mitigation extends quasi-probability ideas to continuous noisy dynamics, where local weak noise can produce global effects during evolution. Combining quasi-probability, symmetry verification, and extrapolation targets different error components and can improve the sampling–estimation trade-off.

  • Continuous-process mitigation: Stochastic error mitigation addresses continuous analog quantum evolution governed by a Lindblad master equation rather than discrete noisy gates.The framework assumes local Lindblad operators with weak strength and also applies to time-dependent Hamiltonians.
  • Continuous-process mitigation: Continuous recovery operations are replaced by Monte Carlo sampling of stronger recoveries because each recovery over a small δt is nearly the identity.This stochastic construction emulates ideal evolution while avoiding the practical challenge of applying recovery operations continuously.
  • Combined mitigation: Symmetry verification removes odd numbers of detectable anti-commuting errors but cannot detect commuting errors or even numbers of anti-commuting errors.Quasi-probability can reduce or transform error components that symmetry verification cannot detect.
  • Combined mitigation: Quasi-probability combined with error extrapolation reduces effective error rates before extrapolation, avoiding the need to adjust the physical error rate but increasing sampling cost.Lower effective error rates can reduce estimation errors compared with naive extrapolation.
  • Combined mitigation: After symmetry verification, the resulting expectation is closer to the noise-free value and approaches the noisy expectation only when μ is large.This comparison is expressed through the symmetry-verified expectation value and the factor 1/cosh μ.
  • Combined mitigation: Hyperbolic extrapolation uses expectation values from different symmetry outcomes to estimate the noiseless value without probing multiple error rates.The combined method was numerically demonstrated on an 8-qubit Fermi-Hubbard simulation under Pauli errors.

Appendix A: Derivation of Eq. (11) for variational quantum simulation

The appendix derives measurement circuits used in variational quantum simulation, including Hadamard tests for unitary expectation values and SWAP-based tests for state overlaps. The destructive SWAP test preserves the overlap evaluation while reducing ancilla and circuit-depth requirements.

  • Variational simulation: McLachlan’s variational principle maps real-time state evolution onto parameter evolution by minimizing the distance between ideal and ansatz-state dynamics.The resulting parameter dynamics support variational quantum simulation.
  • Hadamard test: The Hadamard test evaluates ⟨ψ_in|U|ψ_in⟩ by preparing an ancilla–system state, applying a controlled-U operation, and measuring the ancilla’s Pauli X expectation.Changing the ancilla phase extracts real and imaginary parts, and the same procedure gives Tr[ρ_inU] for mixed inputs.
  • Hadamard test: The Hadamard-test construction also evaluates ⟨ψ_in|V†U|ψ_in⟩ for two unitary operators, supporting coefficient calculations in variational simulation.Shared gates between U and V can reduce the controlled operation to selected circuit components.
  • SWAP tests: The SWAP test evaluates Tr[ρσ] by replacing the Hadamard-test unitary with SWAP and requires 2Nq + 1 qubits.Its gate count can scale as 46Nq + 2 under the cited decomposition.
  • SWAP tests: The destructive SWAP test evaluates the same overlap without an ancilla and with a much shallower circuit using pairwise Bell-basis measurements.Each pair contributes +1 for 00, 01, or 10 and −1 for 11, with the total SWAP result given by their product.

Appendix D: Methodologies for optimisation

The appendix examines optimisation methodologies for variational algorithms, emphasizing local cost functions to address barren plateaus and Hamiltonian morphing to avoid local-minimum difficulties. These methods rely on cost-function structure and sufficiently expressive, finely stepped ansätze.

  • Local cost functions: Local cost functions measure subsets of m < Nq qubits, whereas global cost functions measure all Nq qubits simultaneously.For shallow hardware-efficient ansätze with L = O(log Nq) and m = O(log Nq), local costs can avoid barren plateaus.
  • Local cost functions: The global-cost gradient can decrease exponentially with qubit number, while the local-cost gradient does not exhibit the same barren-plateau behavior.The review states that the barren plateau problem does not occur for the local cost function in the cited setting.
  • Local cost functions: Local and global cost functions share the same zero condition, while their magnitudes satisfy CL ≤ CG ≤ NqCL.A normalized global cost C′G is introduced to handle cases where the norm of |v0⟩ makes costs small.
  • Local cost functions: The local-cost approach alleviated barren plateaus in demonstrations using systems of up to 50 qubits.The cited result concerns trainability of the target quantum circuit.
  • Hamiltonian morphing: Hamiltonian morphing optimisation addresses local-minimum trapping by gradually changing the Hamiltonian, analogously to adiabatic state preparation.When the ansatz is sufficiently powerful and δt is sufficiently small, the procedure is stated to find the global minimum.
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