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Learning-based quantum error mitigation

Armands Strikis, Dayue Qin, Yanzhu Chen, Simon C. Benjamin, Ying Li

arXiv:2005.07601v2quant-ph

TL;DR

NISQ quantum computing needs mitigation methods that do not require a fully known, local error model. This paper learns quasi-probability compensation from classically simulable Clifford variants of the target circuit, demonstrating strong performance on simulated and real systems, including correlated noise.

  • Problem

    Quasi-probability error cancellation depends on sufficiently known and local error models, conditions that are challenging to satisfy for NISQ devices.

  • Method

    The protocol learns an optimal quasi-probability distribution by minimizing errors on Clifford training circuits whose ideal results are efficiently classically simulable.

  • Results

    The protocol performs very well across explored real and virtual quantum systems, including tasks with spatially and temporally correlated noise.

  • Takeaways & Limitations

    The approach offers a practical error-mitigation route that does not require users to access the experimental noise model and can be implemented on current quantum computers.

Abstract

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If NISQ-era quantum computers are to perform useful tasks, they will need to employ powerful error mitigation techniques. Quasi-probability methods can permit perfect error compensation at the cost of additional circuit executions, provided that the nature of the error model is fully understood and sufficiently local both spatially and temporally. Unfortunately these conditions are challenging to satisfy. Here we present a method by which the proper compensation strategy can instead be learned ab initio. Our training process uses multiple variants of the primary circuit where all non-Clifford gates are substituted with gates that are efficient to simulate classically. The process yields a configuration that is near-optimal versus noise in the real system with its non-Clifford gate set. Having presented a range of learning strategies, we demonstrate the power of the technique both with real quantum hardware (IBM devices) and exactly-emulated imperfect quantum computers. The systems suffer a range of noise severities and types, including spatially and temporally correlated variants. In all cases the protocol successfully adapts to the noise and mitigates it to a high degree.

I. INTRODUCTION

The paper develops learning-based quantum error mitigation to avoid reconstructing an experimental error model, using classically simulable Clifford training circuits to optimize compensation. It addresses correlated noise and practical circuit-reconfiguration constraints, with demonstrations on simulated and real quantum systems.

  • Motivation: NISQ devices need error mitigation because conventional quantum error correction may require a prohibitive multiplicative increase in physical qubits.Quantum error correction can suppress errors below threshold, but implementation may require a factor of a thousand or more physical qubits.
  • Learning-based mitigation: The protocol learns the quasi-probability distribution ab initio by minimizing the final computing error on training tasks instead of reconstructing the experimental error model.This learning process automatically incorporates spatial and temporal error correlations.
  • Learning-based mitigation: Clifford-circuit training tasks provide classically evaluable ideal results and are sufficient to find an optimal distribution under negligible single-qubit gate errors.The paper notes that single-qubit gates can achieve substantially higher fidelity than two-qubit gates in some systems.
  • Practical implementation: The learning and mitigated-computation stages support both per-run circuit updates and resource-constrained reconfiguration using Monte Carlo summation or significant-error interventions.The choice depends on whether reconfiguring the quantum circuit is effectively cost-free or costly.
  • Demonstrations: Demonstrations use real hardware and exactly simulated devices with local, spatially correlated, and temporally correlated noise, and the protocol performs very well across explored cases.The experiments include variational quantum algorithm tasks and comparisons with tomography-based methods.
  • Conclusions: The paper evaluates multiple learning strategies and concludes that the approach is simple, effective, flexible, and promising for reliable NISQ-era quantum computing.The strategies vary single- versus multi-parameter forms, summation versus product ansätze, and ideal versus resource-constrained learning.

II. THE GENERAL PROTOCOL

The general protocol inserts adjustable single-qubit Pauli gates around computing gates while keeping noisy multi-qubit frame gates fixed. It estimates an error-free result through a quasi-probability-weighted combination of noisy circuit outcomes and learns the weights using Clifford training circuits.

  • Circuit construction: The circuit fixes Clifford multi-qubit frame gates while treating single-qubit computing gates and inserted Pauli error-mitigating gates as variables.The construction is universal for quantum computing and can be generalized to other circuit configurations.
  • Observable estimation: The protocol defines computation results from measurement outcomes, including observable means such as f(µ) = 1 − 2µ1 for a first-qubit Z measurement.µ is a binary vector representing the measurement outcomes of n qubits.
  • Circuit construction: Figure 1 contrasts circuits without and with mitigation: Pauli layers are added before and after each computing-gate layer, while frame operations remain the error-afflicted components.The example contains four qubits, two frame-gate layers, and three computing-gate layers.
  • Quasi-probability estimation: Error mitigation estimates the error-free computing result by linearly combining results from circuits with different Pauli-gate configurations and quasi-probabilities q(P).The objective is to choose q(P) so that the computing error is minimized.
  • Training: Training uses Clifford computing circuits because their ideal results can be efficiently simulated classically, while noisy results are obtained from the quantum computer.The training set is chosen as a subset T ⊆ C of circuits whose computing gates are Clifford.

III. KEY PROPERTIES

The protocol’s key properties connect an optimal Pauli compensation distribution to the frame-error map and show why Clifford training can generalize to arbitrary computing gates. These results rely primarily on ideal single-qubit gates and a mild transfer-matrix support condition.

  • Existence: An optimal distribution q(P) exists for all computing circuits when single-qubit gates are ideal, independently of the frame-operation error model.With gate-independent single-qubit errors, the proofs continue to hold after adaptation.
  • Exact compensation: Zero mitigation error for all computing circuits requires the error-mitigated frame map Fem to equal the error-free frame map Fef.A solution exists when every nonzero element of Fef has a corresponding nonzero element in the actual error map F.
  • Clifford generalization: The full Clifford training set is sufficient because arbitrary single-qubit unitary maps can be expressed as linear combinations of single-qubit Clifford maps.Optimizing Loss = 0 over all Clifford training circuits therefore yields zero error for Clifford and non-Clifford computing gates under the stated assumptions.
  • Scope: The theoretical properties apply to Pauli, damping, and coherent errors, including both uncorrelated and correlated errors, subject to their stated assumptions.The proofs do not depend on the frame-error model but do assume ideal single-qubit unitary gates in their basic form.

IV. PRACTICAL ISSUES

Direct optimization over all circuits and Pauli configurations grows exponentially, so the paper proposes truncated, factorized, and variational parameterizations. These approaches reduce optimization or sampling costs while retaining practical flexibility.

  • Computational challenge: The spaces of computing circuits and error-mitigating gates grow exponentially with circuit size, making exhaustive optimization impractical.This motivates two practical implementation approaches.
  • Significant-error approach: The significant-error approach truncates the Pauli space to errors expected to matter most, leaving the corresponding quasi-probabilities as optimization parameters.For Pauli errors, the number of parameters usually grows polynomially; numerical evidence later examines training sets three times larger than the significant-error set.
  • Product ansatz: A product-form ansatz assigns independent quasi-probability distributions to significant Pauli errors and represents mitigation by chains of those errors.This avoids optimizing an unrestricted distribution over the full error-mitigating-gate space.
  • Variational ansatz: A variational ansatz parameterizes q(P) with relatively few parameters and can use the full Clifford training set with Monte Carlo loss evaluation.The parameter function may be any real-valued representation, such as a restricted Boltzmann machine.
  • Combined implementations: The three approaches can be combined, and the optimized distribution can be applied using either truncated error-mitigation spaces or Monte Carlo sampling.This allows the implementation to trade circuit-space size, parameterization, and sampling strategy.

V. MULTIPLE OBSERVABLES AND FIDELITY LOSS

The loss function extends from one observable to multiple observables and can alternatively use output-state fidelity, which is estimated through stabilizer measurements for Clifford training circuits.

  • Multiple observables: For N observables, the overall loss is generalized by averaging the individual observable losses.Each f_i(µ) denotes the i-th observable.
  • Fidelity loss: Output-state fidelity provides an observable-independent cost function and can also optimize the quasi-probability distribution.The fidelity compares the ideal final state with the error-mitigated state.
  • Fidelity loss: The fidelity is a pseudo fidelity because the error-mitigated state may not be positive.
  • Fidelity loss: Because training circuits are Clifford, their ideal outputs are stabilizer states, enabling fidelity evaluation through stabilizer-group Pauli measurements.The stabilizer-group summation can be implemented with Monte Carlo sampling.
  • Measurement implementation: Single-qubit Clifford gates inserted before Z-basis measurement change the effective measurement basis without increasing physical circuit complexity when adjacent gates are combined.

VI. PAULI ERROR MODEL

The protocol models noise using spacetime-distributed Pauli errors, obtained by converting general errors through error-mitigating gates and Pauli twirling. Under this model, an inverse quasi-probability map specifies compensating gates that can correct errors for all gate sequences.

  • Pauli error model: Error-mitigating gates and Pauli twirling convert general noise into a symmetrized Pauli error model.Initialization and measurement errors can also be converted into Pauli errors.
  • Pauli error model: Initialization, frame-gate, and measurement errors are represented by n-qubit Pauli operators inserted at defined points in the circuit.Initialization and measurement errors use {I, X}⊗n, while intermediate errors use {I, X, Y, Z}⊗n.
  • Pauli error model: A spacetime error pattern σ combines the Pauli errors across initialization, circuit layers, and measurement, with probability p(σ).
  • Quasi-probability compensation: The inverse noise map assigns quasi-probabilities q(σ), whose solution corrects all errors for every gate sequence R.
  • Quasi-probability compensation: For Pauli noise, error-mitigating gates are needed only in the initialization, frame-gate, and measurement layers; intermediate twirling layers are unnecessary.

VII. SIGNIFICANT-ERROR APPROACH

The significant-error approach makes learning-based mitigation practical by truncating the quasi-probability search to polynomially scaling error sets, then testing it across correlated-noise simulations and variational circuits.

  • Practical parameterisation: The naive quasi-probability optimisation is exponentially large, motivating practical implementations based on truncated significant-error sets.A Pauli-error ansatz can be truncated so the number of optimisation parameters scales polynomially with circuit size.
  • Practical parameterisation: The significant-error approach constructs SigE from locally characterised errors and retains only errors up to a constant order k.The construction begins with tomography-informed error terms and truncates the error-correcting gate set to bounded order.
  • Learning procedure: The truncated loss is evaluated on a randomly selected training subset, and the quasi-probabilities are optimised by least squares before estimating the overhead cost.The subset size comparable to c|SigE| is reported as adequate for learning; mitigation uses qopt(σ) with overhead C = Σσ|q(σ)|.
  • Numerical simulations: The protocol was tested on 8-qubit, 8-layer exact simulations and on correlated spatial and temporal Pauli-noise models, with local noise characterised but correlated noise unknown.The numerical studies compare learning-based mitigation with tomography-based mitigation that uses only the local error model.
  • Numerical simulations: Learning-based mitigation reduced variance and approached the ideal expectation more closely than tomography-based mitigation, while selectively correcting only errors affecting the target observable.The learning process can omit errors that do not influence the computational result, reducing quasi-probability overhead and variance.
  • Variational quantum algorithm: 0.71% versus 3.4%: in the noisy four-qubit variational algorithm, learning-based mitigation achieved a much smaller absolute energy defect than extrapolation.The learning-based estimate reached −8.09 against a true ground-state energy of −8.002, whereas extrapolation reached −7.57.

VIII. PRODUCT-FORM ANSATZ APPROACH

The product-form ansatz parameterizes Pauli error-correction distributions with independently optimized quasi-probabilities and is evaluated under spatially correlated noise. Simulations assess its accuracy and scalability on circuits up to 20 qubits.

  • VIII. PRODUCT-FORM ANSATZ APPROACH: The product-form ansatz represents the error-correcting distribution using optimized quasi-probabilities for Pauli gates.The Pauli configuration is encoded by a binary vector indicating whether each gate is off or on, with normalized quasi-probabilities.
  • Numerical simulations: The simulations use spatially correlated depolarizing noise whose two-qubit error rate depends on circuit size.Additional neighboring two-qubit depolarizing channels are applied with error rate ϵ/10.
  • Numerical simulations: The product-form simulations use circuit layouts with n qubits and N layers, including error-rate parameters ϵ′ specified by circuit size.The circuit-size shorthand is n × N.
  • Numerical simulations: Random single-qubit-unitary circuits benchmark mitigation accuracy through eight qubits and eight layers.Each computing-gate configuration is evaluated using M = 10000 random error-correcting-gate configurations.
  • Average error rescaling factor: Clifford circuits extend the numerical evaluation to circuits of up to 20 qubits because their error-free results can be simulated classically.The average error rescaling factor quantifies mitigation performance.
  • Average error rescaling factor: The error rescaling factor does not increase with circuit size above 9 qubits, indicating efficient scalability of the protocol.Residual error is attributed mainly to finite-sample statistical fluctuations in learning and mitigation.

IX. VARIATIONAL DISTRIBUTION AND MONTE CARLO EVALUATION

The paper generalizes the error-distribution ansatz beyond truncated Pauli models and evaluates it with Monte Carlo sampling. The formulation controls computational overhead and supports scalable loss evaluation.

  • Variational distribution: Variational functions such as restricted Boltzmann machines can represent error models with unknown features using a relatively small parameter set.The quasi-probability distribution is real-valued in this setting.
  • Monte Carlo evaluation: The Monte Carlo loss can use the full Clifford-circuit set, with sampling cost scaling polynomially in summation accuracy regardless of circuit-set size.The section situates this evaluation alongside larger-circuit studies using Clifford computing gates.
  • Variational distribution: The ansatz writes q(P) = CB(P, λ)/A(λ), where B is classically computable and A(λ) normalizes the distribution.When A(λ) cannot be computed directly, samples from |B(P, λ)/A(λ)| can be generated with the Metropolis method.
  • Monte Carlo evaluation: The normalization scale C = Σ_P |q(P)| is the error-mitigation overhead cost.With optimized parameters, Monte Carlo samples of P implement the error-mitigated computation.
  • Monte Carlo evaluation: The Monte Carlo estimator variance is bounded by C^2|f|max^2/M when each sampled circuit is run once.Here M is the number of samples and |f|max is the maximum magnitude of f(μ).
  • Monte Carlo evaluation: The loss is quadratic in C, allowing the optimal C to be found directly for fixed λ while λ is optimized across evaluated loss values.Reusing samples generated at λ′ can reduce sampling requirements.

X. EXPERIMENTAL DEMONSTRATIONS

Experiments on IBM quantum processors apply learning-based mitigation to a two-qubit DQCp circuit and an H2 VQE calculation. The protocol reduces computation error and improves VQE accuracy under hardware noise.

  • Two-qubit DQCp circuit: The protocol improves computation results for a two-qubit DQCp circuit on ibmq 5 yorktown, ibmq ourense, and ibmq santiago.The experiment evaluates ten rotation angles and four Pauli choices for each machine.
  • Two-qubit DQCp circuit: Error-mitigated DQCp results use a learned quasi-probability combination of Pauli-inserted circuits plus a measurement-error term q0.The correction relation is given by Eq. (14).
  • Two-qubit DQCp circuit: Learning uses 24 single-qubit Clifford circuits on each IBMQ machine to obtain optimal q(P) and q0.The required Pauli-inserted results are derived from the same Clifford-gate dataset.
  • Two-qubit DQCp circuit: The DQCp experiment significantly improves accuracy despite no Pauli twirling, with residual errors attributed to statistical fluctuations and non-Pauli errors.Each circuit is run for 8192 shots.
  • Variational quantum eigensolver: The VQE experiment computes the H2 ground-state energy in the minimal STO-3G basis on ibmq santiago with and without learning-based mitigation.The Hamiltonian is partitioned into commuting Pauli groups measured using corresponding circuits.
  • Variational quantum eigensolver: Learning-based mitigation significantly improves the accuracy of the H2 VQE ground-state energy surface on ibmq santiago.Figure 11 compares unmitigated and mitigated results against the Qiskit-computed reference curve.
  • Variational quantum eigensolver: For VQE, Pauli gates are inserted before the single-qubit rotation, with separate circuit modifications evaluating the H1 and H2 Hamiltonian parts.The coefficients are learned separately for each Pauli operator using 24 Clifford gates.

XI. CONCLUSIONS

The paper develops learning-based probabilistic error cancellation using Clifford circuits to learn quasi-probability distributions without reconstructing the full noise model. It establishes information completeness and reports applicability to simulated and IBM quantum hardware.

  • Conclusions: Learning replaces explicit noise-model reconstruction by optimizing a quasi-probability distribution for probabilistic error cancellation.The approach can incorporate spatial and temporal error correlations during training.
  • Conclusions: The protocol was demonstrated with exact simulations of imperfect quantum computers and experiments on IBM quantum devices.The paper reports high performance for circuit sizes comparable to those used in the NISQ era.
  • Conclusions: The learning approach is flexible enough to combine quasi-probability decomposition with noise extrapolation and symmetry-based post-selection.A loss function can optimize the overall strategy when multiple mitigation protocols are combined.
  • Conclusions: The protocol is especially suitable for remote users without access to the experimental noise model and is readily implementable on current quantum computers.Its intended setting is NISQ-era error mitigation.
  • Conclusions: The theoretical construction proves that the relevant Pauli transfer matrices are linearly independent, ensuring that the required solution exists.The information-completeness result extends zero training loss to all unitary gate sequences.

Appendix E: Pauli twirling and error model

This appendix rewrites correlated system-environment noise using Pauli twirling and constructs a tractable set of circuit variations for quasi-probability optimization. The significant-error procedure truncates low-probability variations to keep the optimization polynomially scaling.

  • Pauli twirling: Error-mitigating gates are separated into uniformly sampled Pauli-twirling gates and optimized error-correcting gates.The quasi-probability is set as qc(Pc)/4^n(N+1), making the twirling component uniform.
  • Pauli twirling: Pauli twirling decomposes the effective noisy frame operation into Pauli-error components with nonnegative probabilities summing to one.The construction uses completely positive, trace-preserving noise maps on the environment.
  • Significant-error approach: The significant-error construction first removes circuit variations with zero initial quasi-probability and then excludes the lowest-probability remaining variations.This limits the number of optimized quasi-probabilities to a quantity scaling polynomially with circuit size.
  • Simulation setup: The simulations use eight-qubit, eight-layer circuits with Clifford gates in the training set and Z-basis measurement on the bottom qubit.The layout contains single-qubit unitary gates and controlled-NOT two-qubit gates.

Appendix H: Error model for numerical simulations

The numerical study models local two-qubit noise and adds spatial or temporal correlations separately. It evaluates Pauli-observable estimation on Clifford circuits, where stabilizer structure makes error-free values classically tractable.

  • Base error model: Each controlled-NOT gate is followed by a two-qubit depolarising or dephasing channel with error rate ϵ = 0.01.The base error model acts on the control and target qubits of each controlled-NOT gate.
  • Correlated noise: Spatial correlations are introduced by applying additional equal-rate channels to neighboring qubit pairs with periodic boundary conditions.The correlated channels act on qubits i+1 and i+2, and on qubits i−1 and i, modulo n.
  • Correlated noise: Temporal correlations are modeled by randomly selecting one qubit whose participating channels have increased error rate ϵ* = gϵ, with g = 10 in simulations.The selected qubit follows a uniform probability distribution.
  • Training observables: For Clifford circuits, stabilizer structure determines whether a Pauli observable has mean value 0 or ±1, while Pauli errors can flip nonzero eigenvalues.This enables selection of training circuits with nonzero error-free results.
  • Numerical study: The study uses eight-qubit, eight-layer circuits and tests Clifford overhead constants up to circuit sizes of eight qubits.The data suggest c = 7 is sufficient for small systems, while the simulations use c = 3.

Appendix K: Single-parameter optimisation

This appendix compares single-parameter optimization with multi-parameter learning and details Monte Carlo estimation of the loss and mitigated observables. Single-parameter learning provides only marginal gains over tomography-based mitigation for spatially correlated dephasing.

  • Single-parameter optimisation: Single-parameter learning constrains all quasi-probabilities to depend on one adjustable parameter representing local noise severity.The optimized parameter ϵopt need not equal the severity inferred from two-qubit tomography.
  • Results: Single-parameter learning marginally outperforms its lower bound and is comparable to tomography-based mitigation with k = 2.This result is reported for a 7-qubit, 7-depth circuit with spatially correlated dephasing, and the other error models follow suit.
  • Monte Carlo estimation: The mitigated result is estimated by sampling error-correcting-gate configurations from W(p,b) and averaging weighted single-shot observable outcomes.The sampling distribution is chosen with p = q to minimize variance.
  • Gradient descent: The loss is minimized by computing its gradient with respect to the quasi-probabilities and updating them with a learning rate.A dynamical learning rate is used to keep parameter updates at a reasonable scale.
  • Variational ansatz: The overhead cost of error mitigation is represented by C = Σ_P |q(P)|.A product-form ansatz uses variational parameters λ and C to parameterize the quasi-probability distribution.
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