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Machine Learning for Practical Quantum Error Mitigation

Haoran Liao, Derek S. Wang, Iskandar Sitdikov, Ciro Salcedo, Alireza Seif, Zlatko K. Minev

arXiv:2309.17368v2quant-phcs.LG

TL;DR

Quantum error mitigation can improve noisy quantum results, but its runtime overhead and practical performance across relevant settings remain concerns. This paper benchmarks ML-QEM models and tests their use in hardware experiments and mitigation mimicry. It reports reduced overhead, including more than a 2-fold reduction in a conservative setting.

  • Problem

    Quantum error mitigation must balance accurate results against runtime and sampling overhead, while systematic comparisons across practical tasks and settings have been lacking.

    Source: §I · pp. 1–2

  • Method

    The paper benchmarks four ML-QEM models across circuit classes and noise settings, and tests ML-QEM trained to mimic zero-noise extrapolation for large circuits.

    Source: §I · p. 2, §III · p. 7, Appendix E · p. 16

  • Results

    Across the studied settings, random forests are the best-performing ML-QEM model, and ML-QEM reduces runtime overhead by more than 2-fold in the most conservative setting.

    Source: §I · p. 2

  • Takeaways & Limitations

    ML-QEM can mimic established mitigation methods with lower training and execution overhead, offering a route toward more runtime-efficient mitigation at utility scale.

    Source: §III · p. 7, §5 · p. 20

    The reported empirical error bound applies to in-distribution testing and similar circuits and observables, and depends on the amount of training data.

    Source: §5 · p. 15

Abstract

from arXiv · show

Quantum computers progress toward outperforming classical supercomputers, but quantum errors remain their primary obstacle. The key to overcoming errors on near-term devices has emerged through the field of quantum error mitigation, enabling improved accuracy at the cost of additional run time. Here, through experiments on state-of-the-art quantum computers using up to 100 qubits, we demonstrate that without sacrificing accuracy machine learning for quantum error mitigation (ML-QEM) drastically reduces the cost of mitigation. We benchmark ML-QEM using a variety of machine learning models -- linear regression, random forests, multi-layer perceptrons, and graph neural networks -- on diverse classes of quantum circuits, over increasingly complex device-noise profiles, under interpolation and extrapolation, and in both numerics and experiments. These tests employ the popular digital zero-noise extrapolation method as an added reference. Finally, we propose a path toward scalable mitigation by using ML-QEM to mimic traditional mitigation methods with superior runtime efficiency. Our results show that classical machine learning can extend the reach and practicality of quantum error mitigation by reducing its overheads and highlight its broader potential for practical quantum computations.

I. INTRODUCTION

Quantum error mitigation can reduce errors, but its runtime overhead and uncertain performance across practical settings remain obstacles. The paper benchmarks ML-QEM across models and circuit settings, reporting lower mitigation overhead and experiments on circuits up to 100 qubits.

I. INTRODUCTION

  • Quantum error mitigation seeks near-noise-free results despite unmonitored device errors, but practical mitigation must balance accuracy against runtime overhead and potentially exponential sampling costs.The passage contrasts QEM’s algorithmic runtime costs with quantum error correction’s qubit and real-time monitoring overheads.Source: §I · p. 1
  • The paper identifies whether ML-QEM works across device noise profiles, circuit classes, and large circuits beyond classical simulation as an open practical question.It also notes the lack of systematic, equal-footing comparisons of traditional methods and statistical models across relevant tasks.Source: §I · pp. 1–2
  • The study compares linear regression, random forests, multi-layer perceptrons, and graph neural networks across circuit classes, noise profiles, and application scenarios.It includes random circuits and Trotterized Ising dynamics, as well as unseen-observable mitigation and variational quantum-classical tasks.Source: §I · p. 2
  • In the most conservative setting, ML-QEM reduces runtime overhead by more than 2-fold, with experiments reaching 100 qubits and two-qubit gate depth 40.The authors frame ML-QEM as a way to mimic traditional mitigation methods with superior runtime efficiency.Source: §I · p. 2

II. RESULTS

  • For intractable circuits, ML-QEM can be trained against outputs from a conventional mitigation strategy and then predict mitigated expectation values directly from noisy QPU data.This runtime workflow avoids additional mitigation circuits after training.Source: §II · pp. 2–3, Figure 1 · p. 2

A. Performance Comparison at Tractable Scales

At tractable scales, the study compares ML-QEM models with digital zero-noise extrapolation on random and Trotterized circuits under different noise conditions. Random forests perform strongly in simulations and hardware tests, while added coherent errors constrain extrapolation performance.

A. Performance Comparison at Tractable Scales

  • The simulations cover four-qubit random circuits with two-qubit-gate depths up to 18 and Trotterized Ising circuits tested both within and beyond the training Trotter-step range.Trotterized tests also vary coupling strengths, use randomly chosen Pauli bases for weight-one observables, and examine readout and coherent noise in addition to incoherent noise.Source: §1 · p. 3, A. Performance Comparison at Tractable… · p. 3, §2 · p. 4
  • For random circuits, all four ML-QEM models are competitive with digital ZNE despite ZNE having twice the runtime overhead, and random forests consistently outperform the other ML models.The evaluation uses four-qubit random circuits simulated with FakeLima noise.Source: §1 · pp. 3–4, Figure 2 · p. 3
  • With incoherent gate noise, random-forest ML-QEM performs better than ZNE with or without readout error, and random forests lead the ML models in interpolation and extrapolation.The reported model ranking is random forests, followed closely by MLP, OLS, and GNN.Source: §2 · p. 4
  • Under added coherent CNOT over-rotations averaging 0.04π, none of the models effectively mitigate errors in extrapolation.The authors suggest suppressing coherent errors before applying ML-QEM in this setting.Source: §2 · p. 4
  • On IBM hardware, random forests outperform ZNE at all tested Trotter steps, with overhead reductions of up to 40% overall and 50% at runtime.The experiment used 50 training circuits per step and tested on 250 circuits per step.Source: §2 · p. 4

B. Mitigating Unseen Pauli Observables

ML-QEM is tested on unseen Pauli observables and in VQE, where the reported results show mitigation can outperform ZNE while reducing the need for additional mitigation circuits.

B. Mitigating Unseen Pauli Observables

  • Tomography and VQE measure multiple non-commuting Pauli observables on the same gate sequence using different measurement bases, creating multiple target circuits.The authors propose ML-QEM as a way to improve mitigation performance with lower overhead in these settings.Source: B. Mitigating Unseen Pauli Observables · p. 4
  • Training on ≲2% of 4,095 Pauli observables yielded lower errors on unseen observables than ZNE.The RF model was trained on a six-qubit, five-step Trotterized Ising circuit.Source: B. Mitigating Unseen Pauli Observables · pp. 4–5

C. Enhancing Variational Algorithms

  • ML-QEM was trained beforehand to mitigate VQE energy estimates, avoiding additional mitigation circuits at each optimization iteration.Traditional mitigation can require extra circuits during every iteration.Source: C. Enhancing Variational Algorithms · p. 5
  • For the H2 dissociation curve, RF produced energies close to chemical accuracy and smaller absolute errors than ZNE-mitigated energies.The model was trained on 2,000 circuits with parameters sampled from [−5, 5]; no readout error mitigation was applied.Source: C. Enhancing Variational Algorithms · p. 5

D. Scalability through Mimicry

The paper uses ML-QEM to mimic ZNE on 100-qubit hardware circuits, reporting similar mitigation behavior with reduced quantum-resource overhead. The discussion frames mimicry as a way to accelerate conventional methods when ideal outcomes are classically intractable.

D. Scalability through Mimicry

  • RF accurately mimicked ZNE on 100-qubit Trotterized TFIM circuits, including non-Clifford tests where ideal expectations are classically intractable.The experiment used ZNE with TREX and ran on IBM’s ibm brisbane QPU.Source: D. Scalability through Mimicry · p. 6
  • Mimicry reduced overall quantum resources by 25% and runtime overhead by 50% compared with ZNE.The approach trains ML-QEM to imitate another QEM method rather than outperform the method being mimicked.Source: D. Scalability through Mimicry · p. 6

III. DISCUSSION

  • The study reports that RF was the best-performing model overall and that ML-QEM sometimes outperformed ZNE across varied circuits, noise, and applications.It also notes that mimicry provides similar performance with lower overall overhead, extending to classically intractable circuits.Source: §III · p. 6
  • At utility scale, ML-QEM is presented as an accelerator for conventional state-of-the-art mitigation methods, not as a competing QEM technique.The proposed focus is to mimic their quality with less training and execution overhead.Source: §III · p. 7

IV. METHODS

The supplied Methods passages emphasize model-selection dependencies and illustrate the 100-qubit RF-ZNE mimicry experiment.

A. Statistical Learning Models

  • The Methods section introduces discussion of statistical models, data encoding strategies, and hyperparameters.The supplied passage does not specify those strategies or settings.Source: A. Statistical Learning Models · p. 7
  • On 100-qubit ibm brisbane circuits, RF-ZNE mimicked ZNE with 25% lower overall and 50% lower runtime quantum-resource overhead.The reported training included 10 circuits per Trotter step, with 40 test circuits per step.Source: Figure 6 · p. 8
  • Model performance depends on training-set size, encoding scheme, architecture, hyperparameters, and the task.The authors describe their models as a starting point for practitioners choosing an approach for an application.Source: A. Statistical Learning Models · p. 8

1. Linear Regression

Linear regression predicts ideal expectation values from circuit features and noisy measurements, while random forests average predictions from an ensemble of decision trees. The section contrasts ML-QEM’s broad circuit-and-observable scope with CDR’s single-circuit, single-observable approach.

1. Linear Regression

  • CDR fits a circuit- and observable-specific regression using simulated and hardware expectation values from near-Clifford circuits that mirror the target circuit.Its approach retains a small number of non-Clifford gates as control knobs, while keeping their number low to manage classical simulation costs.Source: §1 · p. 8
  • ML-QEM targets large classes of circuits, observables, and parameters, whereas CDR [16, 17, 31] focuses on one circuit and one observable.For the many-body experiments, the target batch typically includes all weight-one and weight-two Pauli observables and scales polynomially with system size.Source: §1 · p. 8
  • Linear regression maps circuit features and noisy expectation values to ideal values through a fitted linear function.The inputs include native-gate counts, a sparse Pauli representation of the observable, and optional device-noise parameters; the experiments use ordinary least squares.Source: §1 · pp. 8–9, Figure 7 · p. 9

2. Random Forest

  • Random forests average predictions from an ensemble of decision trees, with bootstrap aggregation and random feature subsets used to improve prediction accuracy and reduce overfitting.Their input features match those used for OLS.Source: §2 · p. 9

3. Multi-layer Perceptron

The section describes an MLP using fully connected layers and a GNN that processes circuit graphs through message passing. It also specifies the digital zero-noise extrapolation reference used in the study.

3. Multi-layer Perceptron

  • The MLP uses fully connected layers with nonlinear activations to model nonlinear relationships, and the study reports competitive mitigation performance while noting susceptibility to overfitting.The experiments use two dense layers, hidden size 64, and ReLU activation.Source: §3 · p. 10

4. Graph Neural Network

  • The GNN encodes a transpiled circuit as an undirected acyclic graph, then uses message passing and dense layers to predict mitigated expectation values.Node features include gate type, gate errors, and relevant qubit coherence and readout errors; additional crosstalk and idling features were not helpful for the problems studied.Source: §4 · p. 10
  • The authors suggest investigating whether setting encoded GNN noise parameters to a low-noise regime could support extrapolation toward near-noise-free expectation values without ideal-value training.They frame this as a potential direction for future investigation on large circuit datasets.Source: §4 · p. 10

B. Zero-noise Extrapolation

Appendix A: Depolarizing Noise

Under successive depolarizing channels, noisy and ideal expectation values are linearly related for the described Trotterized circuits. The appendix also gives a train-test break-even condition for ML-QEM mimicry of traditional mitigation.

Appendix A: Depolarizing Noise

  • Successive depolarizing channels combine into a depolarizing channel with an effective rate p(l), yielding a linear dependence between noisy and ideal expectation values.The appendix derives the combined channel by induction across the l circuit layers.Source: Appendix A · p. 12
  • For fixed-step, fixed-brickwork Trotter circuits with a fixed number of depolarizing-noise layers, ML-QEM’s linear regression is unbiased, while linear ZNE remains biased.The noisy and ideal expectation values lie on a line, but noise amplification gives ZNE estimates with differently amplified noise that lie on different lines.Source: Appendix A · p. 12

Appendix B: Break-even in Overall Overhead of ML-QEM

  • ML-QEM mimicry has higher overall efficiency when n_train/n_test is below (m − 1)/m, where m is the number of executions required by the mimicked QEM method.The comparison equates traditional runtime cost m n_test with ML cost m n_train + n_test.Source: Appendix B · p. 12

Appendix C: Additional Experimental Details

The appendix describes simulation noise settings and experimental sampling, then reports random-circuit and Trotterized-Ising findings. In Trotter tests, random forests performed best among ML-QEM models under incoherent noise and in interpolation with added coherent noise.

Appendix C: Additional Experimental Details

  • Simulation and experimental non-ideal expectation values were obtained from measurement statistics using 10,000 shots.Source: Appendix C · p. 12

1. Errors Used in Simulations

  • The simulation error parameters represent typical-to-high-end device error strengths, and the table reports average gate infidelity averaged over five qubits per fake backend.Coherent CNOT error was added in simulations to investigate performance under relatively strong coherent errors.Source: §1 · p. 12, Table I · p. 13

2. Random Circuits

  • For random circuits, the study reports that rigorous hyperparameter optimization could affect the methods’ relative performance and leaves that analysis to future work.Source: §2 · p. 13

3. Trotterized 1D Transverse-field Ising Model

  • In the Trotter simulations, added coherent noise made ML-QEM and digital ZNE deteriorate, and none of the models mitigated effectively in extrapolation.The expanded Fig. 8 averages depth-dependent performance over 9,000 Ising circuits and reports poor extrapolation with additional coherent noise.Source: Figure 8 · p. 13
  • In Trotterized 1D TFIM tests, random forests performed best among ML-QEM models in interpolation and extrapolation with incoherent noise and readout error, and in interpolation with added coherent noise.For real-hardware experiments on ibm algiers, RF outperformed ZNE with maximal p-value 3.1 × 10−10; ML-QEM used 3,000 circuits versus ZNE’s 5,000, a 40% overall overhead reduction and 50% runtime overhead reduction.Source: §3 · p. 14
  • In the small-scale Trotter study, noisy expectation values correlated with ideal values, while gate-count correlations were weaker and suggested that models used depth information in mitigation.Source: §3 · p. 14

4. Enhancing Variational Algorithms

For variational algorithms, ML-QEM can reduce training needs by generalizing across Pauli observables, and RF mimicry of ZNE cuts circuit and runtime costs. Under drifted noise, MLP fine-tuning and RF retraining both converge with relatively few new circuits.

4. Enhancing Variational Algorithms

  • Training on a subset of Hamiltonian Pauli observables lets RF generalize to the remaining observables in the H2 VQE task.The RF was trained on X1X2 and Z1Z2 measurements, while the Hamiltonian also includes I1Z2 and Z1I2.Source: §4 · p. 14
  • With readout errors removed, noisy expectation values are closer to ideal, and RF and ZNE perform comparably on the remaining errors.The simulations are described as essentially equivalent to applying readout-error mitigation such as TREX.Source: §4 · p. 14, §5 · p. 15

5. Scalability through Mimicry

  • 25% fewer total quantum resources and 50% lower runtime are reported for RF-mimicking-ZNE than traditional ZNE.The comparison uses two noise factors for ZNE; RF-mimicking-ZNE uses 100 training circuits and 400 test circuits.Source: §5 · pp. 14–15
  • For similar circuits and observables, the RF mimic model's mitigation error is reported as upper-bounded by approximately 0.1, subject to the training data used.The authors say this accuracy can be expected with high probability when reusing the same RF model on similar circuits and observables.Source: §5 · p. 15

Appendix D: Efficient Adaptability to Drifted Noise

  • On FakeBelem, fine-tuning an MLP trained on FakeLima converges after about 300 circuits, while training that MLP from scratch converges after about 500.Both MLP approaches reach the same testing performance; an RF trained from scratch converges in fewer than about 300 circuits.Source: Appendix D · p. 15
  • As device noise drifts over time, an ML-QEM model can be fine-tuned for a different noise model and achieve similar performance with substantially less training data; MLPs adapt efficiently, while RFs train from scratch as efficiently.Source: Appendix D · pp. 15–16

Appendix E: Comparison and Resource Analysis of QEM Methods

The appendix compares QEM methods by precision and resource costs, focusing on how ML-QEM can mimic established methods. It describes PEC's learning and sampling costs, and concludes that PEC is impractical for the paper's large, deep circuits.

Appendix E: Comparison and Resource Analysis of QEM Methods

  • ML-QEM can mimic a physics-based QEM method, and its expected runtime benefit depends on the method being mimicked.The comparison rates accuracy, precision, scalability, training efficiency, and mitigation efficiency; ML-QEM mimicry receives ratings similar to its target method.Source: Appendix E · pp. 16–17, Table II · p. 17

1. Probabilistic Error Cancellation (PEC)

  • PEC learns a noise model for circuit layers using Pauli twirling and amplified learning experiments, then uses that model for mitigation.The learning cost grows with device connectivity and linearly with the number of layers.Source: §1 · pp. 17–18
  • PEC learning has a precision constraint: a no-go theorem permits only products of certain Pauli coefficient pairs to be determined precisely.Breaking the degeneracy introduces susceptibility to SPAM and some additive errors, although the protocol appears to work at current experimental scales.Source: §1 · p. 18
  • The PEC runtime analysis assumes error-free noise-model learning and specifies a maximum allowable error between ideal and mitigated expectation values.The generalized per-layer, per-qubit sampling overhead reflects device quality; current superconducting-device values are stated as 1.01 to 1.04.Source: §1 · p. 18
  • PEC's runtime rises exponentially with circuit area, making it impractical for the large, deep circuits in the experiments; the authors therefore use ZNE.The runtime estimate combines sampling overhead and circuit-layer operations per second; Fig. 11 illustrates the increase.Source: §1 · pp. 17–18, Figure 11 · p. 19

2. Zero-noise Extrapolation (ZNE)

ZNE estimates noise-free expectation values by extrapolating measurements from circuits run at amplified noise levels. Its simplicity and use in experiments come with an assumption about predictable noise scaling and less precise sampling-cost estimates.

2. Zero-noise Extrapolation (ZNE)

  • ZNE extrapolates expectation values measured at amplified noise levels to the zero-noise limit, assuming a relationship between amplification and observed values.The relationship is commonly modeled as linear, polynomial, or exponential based on heuristic or empirical considerations.Source: §2 · p. 19
  • Unlike PEC, ZNE requires no explicit noise characterization or calibration, but is not considered bias free.Its use assumes noise effects scale predictably with the amplified noise level.Source: §2 · p. 19
  • ZNE does not inherently provide tight sampling-cost bounds; its cost is anticipated to scale exponentially with the number of amplification factors.The cost also depends on calibrating the amplification factors and sampling increasingly damped signals.Source: §2 · p. 19

3. Probabilistic Error Amplification (PEA)

The section describes probabilistic error amplification as a calibrated way to scale noise for ZNE and contrasts ML-QEM’s broad scope with CDR’s circuit-specific training. Its summary highlights utility-scale experiments applying ML-QEM to mimic ZNE and the potential to improve mitigation runtime efficiency.

3. Probabilistic Error Amplification (PEA)

  • PEA combines ZNE with PEC noise learning and error injection to amplify noise in a more controlled, calibrated manner.The noise scale is set by scaling the Lindblad generator; quasi-probability sampling is used for α < 0 and probability sampling for α > 0.Source: §3 · p. 20

4. Clifford Data Regression (CDR)

  • CDR replaces most non-Clifford gates with Clifford gates to generate near-Clifford training circuits for a linear model linking ideal and noisy expectations.Keeping non-Clifford gates ideally below fifty supports practical classical simulation, but applying CDR across circuits or observables requires separate training data and creates conflicting design demands.Source: §4 · p. 20

5. Summary

  • The section compares learning and mitigation runtime overheads for ZNE, PEC, and PEA, and also discusses QSE and CDR.Tab. II summarizes qualitative behaviors of the black-box methods.Source: §5 · p. 20
  • Experimental data showcase a utility-scale application of ML-QEM to ZNE mimicking, alongside analyses of PEC’s limitations and sampling-overhead challenges.The authors see potential for ML-QEM to improve the runtime efficiency of established and forthcoming physics-based mitigation strategies.Source: §5 · p. 20

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