Error mitigation with Clifford quantum-circuit data

Piotr Czarnik; Andrew Arrasmith; Patrick J. Coles; Lukasz Cincio

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Error mitigation with Clifford quantum-circuit data

Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, Lukasz Cincio

arXiv:2005.10189v3May 20, 2020quant-ph

TL;DR

Near-term quantum advantage requires accurate observables despite hardware noise, motivating scalable error mitigation. The paper introduces Clifford Data Regression, which learns corrections from largely Clifford training circuits that are classically simulable. It reports order-of-magnitude error reductions for 16-qubit IBMQ and 64-qubit noisy-simulator ground-state-energy problems.

Problem
NISQ hardware lacks fault-tolerant error correction and therefore provides observables with limited accuracy, while practical applications require accurate estimates.
Method
Clifford Data Regression fits an ansatz to noisy quantum-computer and exact classical observables generated from largely Clifford, classically simulable circuits.
Results
An order-of-magnitude error reduction was obtained for a 16-qubit IBMQ ground-state-energy problem and a 64-qubit noisy-simulator problem.
Takeaways & Limitations
CDR provides meaningful corrections for a 64-qubit ground-state-energy problem and can mitigate deeper circuits with modest hardware-noise reductions.
Takeaways & Limitations
Near-Clifford simulation cost grows exponentially with the number of non-Clifford gates, and further testing is needed for real hardware, scaling, and non-variational training sets.

Abstract

from arXiv · show

Achieving near-term quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable error-mitigation method that applies to gate-based quantum computers. The method generates training data $\{X_i^{\text{noisy}},X_i^{\text{exact}}\}$ via quantum circuits composed largely of Clifford gates, which can be efficiently simulated classically, where $X_i^{\text{noisy}}$ and $X_i^{\text{exact}}$ are noisy and noiseless observables respectively. Fitting a linear ansatz to this data then allows for the prediction of noise-free observables for arbitrary circuits. We analyze the performance of our method versus the number of qubits, circuit depth, and number of non-Clifford gates. We obtain an order-of-magnitude error reduction for a ground-state energy problem on 16 qubits in an IBMQ quantum computer and on a 64-qubit noisy simulator.

1 Introduction

Near-term quantum advantage requires accurate observables despite NISQ noise, but scalable error mitigation remains difficult. This work proposes a Clifford-based method that learns corrections from classically simulable training circuits and reduces errors substantially in tested problems.

Motivation: NISQ devices lack sufficient qubits and gate fidelities for fault-tolerant correction, limiting observable accuracy.Chemical and optimization applications require high-accuracy estimates to outperform classical methods.
Existing approaches: Error mitigation reduces noise impact without directly correcting it, commonly through classical post-processing or circuit and algorithm design.Zero-noise extrapolation uses measurements at multiple noise levels to estimate the zero-noise expectation value.
Motivation: Scalability is crucial because methods effective on small systems may not work effectively beyond 50 qubits or scale well in practice.The paper frames quantum-supremacy-scale error mitigation as substantially more challenging than small-system mitigation.
Contribution: The proposed method generates noisy quantum-computer data and noiseless classical data from largely Clifford circuits, fits an ansatz, and predicts noise-free observables.Clifford circuits are efficiently classically simulable, enabling training-data construction.
Results: An order-of-magnitude error reduction was obtained for 16-qubit IBMQ and 64-qubit noisy-simulator ground-state-energy problems.The method was also demonstrated for quantum phase estimation and appeared to outperform ZNE on larger-scale problems considered.

2 Our method

Clifford Data Regression constructs classically tractable training states, pairs noisy quantum measurements with exact classical values, and fits a model to predict noise-free observables. Training-state selection can be tailored to the target, while accuracy and cost depend on refinement and simulation resources.

Method: CDR targets an observable for a state |ψ⟩ by learning from noisy and exact expectation-value pairs.The noisy value comes from the quantum computer, while the exact value is computed classically.
Training data: Training states are constructed from circuits composed largely of Clifford gates, with N non-Clifford gates serving as a refinement parameter.Each training state must permit efficient classical computation of the observable expectation value.
Model fitting: The method fits an ansatz for the noise-free observable using regression or machine learning; this work uses least-squares linear regression.The fitted model is then applied to predict the target observable.
Training-state selection: Training states can be biased toward the target state using nearby Clifford substitutions or MCMC sampling.For variational algorithms, MCMC can sample classically simulable states near cost-function extrema and performs better for deep circuits.
Cost and performance: For QAOA, 60–80 training circuits and 8192 shots per expectation estimate produced an order-of-magnitude improvement across Q = 8–64 and p = 1–24.The required circuit count and shot precision must be determined empirically from convergence.
Computational cost: Near-Clifford expectation-value simulation grows exponentially with N and polynomially with the number of Clifford gates.The authors used N ≤30 for their proof of principle, although state-of-the-art simulators can simulate N = 80.

3 Numerical Implementations

The numerical implementations apply CDR to QAOA ground-state energy optimization and quantum phase estimation, examining performance across circuit depth, qubit count, and non-Clifford-gate count. CDR reduces errors substantially, including order-of-magnitude improvements for 16-qubit hardware, 64-qubit simulations, and deep circuits.

QAOA implementation: CDR corrects low-energy local minima of the transverse Ising model using QAOA circuits and near-Clifford training circuits.The QAOA ansatz uses variational parameters β_j and γ_j, with p denoting the number of layers; training circuits contain N non-Clifford gates.
QAOA implementation: 0.409 noisy mean relative energy error falls to 0.028 after correction for the 16-qubit IBM Almaden experiment.The experiment used p = 2 QAOA rounds, 16384 shots per circuit, and 63 near-Clifford training circuits.
Scaling with N, p, and Q: One to two orders of magnitude of energy-error reduction is obtained across variations in N, p, and Q.Increasing N monotonically reduces error because the training set becomes closer to the target state, while larger depth and qubit count make correction more challenging.
Deep QAOA circuits: Order-of-magnitude improvement persists at p = 24 in an eight-qubit noisy simulator with noise rates reduced by a factor of 6 relative to the p = 4 model.The deepest circuits contained 336 CNOTs and 360 non-Clifford rotations, with N = 28 in the training circuits.

4 Conclusions

CDR uses classically simulable, largely Clifford circuits to learn corrections for noisy observables, enabling meaningful mitigation at larger problem sizes. The authors report substantial error reduction but identify further hardware, scaling, and benchmarking work as necessary.

CDR learns to correct noisy observables using training data generated from circuits composed largely of Clifford gates.Such circuits are efficiently classically simulable.
A 64-qubit ground-state-energy problem received a meaningful CDR correction with a shot count feasible for current quantum devices.
Further real-hardware testing, scaling studies, and benchmarks are needed, especially for non-variational algorithms.

A Motivation for linear ansatz

The paper motivates a linear correction ansatz by showing that global depolarizing noise creates an exact linear relation between noisy and noiseless expectation values. Under related measurement-noise models, the same ansatz can also exactly correct the noise, although it is not expected to cover all noise types.

The linear ansatz is theoretically motivated because global depolarizing noise produces a linear relation between noisy and noise-free expectation values.
For global depolarizing noise, linear regression fits the training data perfectly and recovers the exact correction parameters.
The same linear ansatz can perfectly correct certain symmetric or asymmetric measurement-noise channels.
The authors do not expect the linear ansatz to correct all noise types and leave more complicated ansatzes for future study.

B Generating the set Sψ

CDR constructs its training set by sampling near-Clifford circuits with MCMC, preserving a fixed number of non-Clifford gates while biasing circuits toward relevant energy regions. The resulting classical cost is observed to scale polynomially with qubit number and QAOA depth for fixed non-Clifford count, with deviations at the largest depths.

MCMC generates near-Clifford training circuits by proposing local gate substitutions and accepting or rejecting each update with a Metropolis-Hastings rule.The update preserves the number of non-Clifford gates N.
The QAOA likelihood is chosen to bias training circuits toward the ground-state energy, while an alternative monotonic likelihood achieves similar mitigation quality without solution knowledge.
For fixed N, classical resources depend on MCMC autocorrelation, initialization length, and the number of Clifford gates.
The total classical simulation cost is observed to scale polynomially with Q and p for fixed N, although the largest depths show deviations.

C Error bars

CDR error bars primarily quantify imperfect fitting of the training data, while mismatch between training and target-state noise is a separate and difficult-to-assess error source.

CDR error bars reflect confidence in the predicted noise-free observable rather than a complete accounting of all error sources.
The two principal error sources are imperfect training and the inability of training data to capture noise affecting the target state.
The reported error bars focus on imperfect training because target-state noise mismatch is difficult to quantify.
The error-bar magnitude is based on three standard deviations, with the standard deviation derived from C/(L − 1).

D QAOA circuit structure

The QAOA and quantum phase estimation circuits are composed almost entirely of Clifford gates, with non-Clifford σZ rotations providing the only exceptions and training-data variations.

QAOA circuits contain (2Q − 2)p CNOTs and (2Q − 1)p non-Clifford RZ gates, with all other gates being Clifford.
The QAOA training circuits are generated by replacing selected σZ rotations with Sn while leaving the remaining circuit structure unchanged.
The phase estimation circuit estimates Re(⟨χ|e−i ˜Ht|χ⟩) for a randomly chosen product state using compiled parameters.
The quantum phase estimation circuit likewise has only σZ rotations as non-Clifford gates, with training data generated by replacing some of them with Sn.

F Zero Noise Extrapolation

Zero noise extrapolation stretches circuit evolution while reducing drive amplitude to emulate stronger noise, then extrapolates observables toward the noiseless limit under stated noise-model assumptions.

ZNE approximates noiseless computation by extrapolating observables toward the limit λ → 0, where λ represents noise-channel strength.
Under time-rescaling invariance of L and independence from K(t), stretching evolution time while reducing drive magnitude effectively increases λ.
The stretched evolution uses a modified drive Hamiltonian and a rescaled state ρ′(t) = ρ(c−1t) over the longer duration.
Under the stated assumptions, evolving longer with the stretched drive is equivalent to original evolution with λ → cλ.

F.2 Implementation

The implementation uses selected circuit compilations, pulse stretching, measurement-error mitigation, and extensive sampling to evaluate the method experimentally.

The implementation uses stretch factors c ∈ {1, 1.1, 1.25, 1.5} for the QAOA pulse sequences.
For quantum phase estimation, parameters f1(t) through f6(t) are obtained by compiling the circuit accurately up to numerical precision.
The phase-estimation circuit uses a randomly chosen product state and assumes all-to-all device connectivity.
For Q = 16 and p = 2, stretching lowers pulse-envelope amplitudes while increasing their durations.
Each QAOA operator is measured with 212992 shots at every stretch factor, alongside built-in measurement-error mitigation.

F.3 Results

The zero-noise-extrapolation results compare several fitting forms across three low-energy QAOA instances, but the method does not accurately correct the energy expectations in this use case.

Three low-energy QAOA instances are evaluated using linear, quadratic, cubic-polynomial, and exponential extrapolation fits.
For this particular use case, ZNE did not provide an accurate correction for the energy expectation values.

G Correction with a constant ansatz

The paper examines whether CDR can remain effective when exact training observables are unavailable, comparing it with a constant correction under realistic IBM Ourense noise. CDR outperforms the constant ansatz, indicating that training-set construction need not provide exact observables in this test.

Training-set construction: Training circuits are selected to resemble the circuit of interest, using similar energy for QAOA and similar noisy expectation values for Quantum Phase Estimation.This motivates examining whether correction quality depends on similarity between exact training and target observables.
Training-set construction: For global depolarizing noise, CDR learns the perfect correction from any two training circuits with different observable values.The result provides an analytic case where substantially different training observables can still suffice.
Comparison with constant correction: The analysis compares CDR with a constant ansatz for the more realistic IBM Ourense noise model.Average absolute fitted slope and intercept values are reported across QAOA minima and energy-contributing observables.
Comparison with constant correction: CDR outperforms constant correction for p = 3 QAOA circuits under the IBM Ourense noise model.The comparison uses the same training sets and local minima as the corresponding CDR experiments.
Comparison with constant correction: The test uses MCMC-constructed training sets previously used for CDR correction in the QAOA experiments.These sets are evaluated on p = 3 QAOA circuits corresponding to Figs. 3(c) and 8.