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Weakly Supervised Quantum Error Mitigation

Seyed Mohamad Ali Tousi, G. N. DeSouza

arXiv:2609.25555v1cs.AI

TL;DR

Supervised mitigation depends on ideal outputs from noiseless simulation, whose cost grows exponentially with system size. This paper uses weak signals from circuit structure and hardware to predict per-qubit error rates and apply readout-channel inversion. It reports stronger mitigation than published analytical baselines on two IBM devices, while supervised models remain stronger where ideal labels are available.

  • Problem

    Supervised mitigation requires ideal output distributions from noiseless simulation, whose cost grows exponentially with system size.

    Source: §1 · p. 1

  • Method

    Sixteen labeling functions use circuit and hardware signals to produce soft per-qubit error probabilities, which are generalized to unseen circuits and used as readout-channel rates.

    Source: §1 · p. 1, §2.5 · p. 3, §4.2 · p. 4

  • Results

    The method outperforms published analytical baselines on two IBM devices, while fully supervised models remain stronger where ideal labels exist.

    Source: §1 · p. 2, §8 · p. 14

  • Takeaways & Limitations

    Weak supervision provides a mitigation approach for settings where ideal training labels cannot be computed, with a measurable but modest cost relative to true labels.

    Source: §8 · p. 14

    The label model requires a small validation slice derived from ideal outputs for recalibration, so ideal-label use is reduced but not eliminated.

    Source: §7.4 · p. 13

Abstract

from arXiv · show

Supervised approaches to quantum error mitigation learn a map from noisy circuit outputs to ideal ones, and therefore require the ideal outputs. Producing those ideal outputs demands noiseless classical simulation, whose cost grows exponentially with system size, so supervision is unavailable in exactly the regime where mitigation matters most. We ask whether cheap, individually unreliable signals drawn from circuit structure and hardware calibration can take the place of ideal labels. We assemble sixteen heuristic labeling functions (stabilizer and parity constraints, relaxation and readout characteristics, local depth, gate counts, and neighboring activity), reconcile their disagreements with a probabilistic label model, and read the resulting per-qubit error probabilities as a readout channel whose inverse mitigates the measured distribution. No ideal output enters the training path. On $147{,}000$ five-qubit circuits executed on two IBM devices, the method removes $24.3\%$ (Algiers) and $28.8\%$ (Hanoi) of the Kullback-Leibler divergence to the ideal distribution, against $15.4\%$ and $21.5\%$ for the strongest published analytical baseline, a margin that holds on both devices and lies far outside its bootstrap interval. Supervised neural models trained on ideal distributions remain stronger where such labels exist, and we quantify that gap rather than setting it aside; the method's claim is to the regime where they do not, since the labels they require cannot be computed for the circuits mitigation is needed for. The codes will be released shortly.

1 Introduction

Quantum error mitigation can recover ideal measurement statistics without error-correction overhead, but supervised mitigation depends on noiseless simulation that becomes exponentially costly as system size grows. This paper replaces ideal training labels with weak signals from circuit structure and hardware, then uses their inferred error probabilities to mitigate measurements.

1 Introduction

  • Ideal labels for supervised mitigation require noiseless simulation whose cost grows exponentially with qubit count, limiting support where mitigation is most needed.The ideal distribution is needed to train a map from noisy to ideal output distributions.Source: §1 · p. 1
  • Sixteen heterogeneous labeling functions vote on per-qubit corruption without consulting ideal outputs.The functions use information available to practitioners, including calibration, circuit structure, and parity or stabilizer constraints.Source: §1 · pp. 1–2
  • Predicted per-qubit error probabilities define a tensor-product readout channel that is inverted to mitigate the measured distribution.This places the learned method and classical baselines on common correction machinery, with rates supplied from different sources.Source: §1 · p. 2
  • The evaluation compares the method with published analytical baselines on two IBM devices and separately quantifies its gap to fully supervised models.The study uses bootstrap intervals, multiple seeds, and paired tests.Source: §1 · p. 2

2 Related Work

The paper connects quantum error mitigation with programmatic weak supervision, contrasting analytic corrections and learned methods by their noise assumptions and sources of training signal. Its approach predicts circuit-specific readout rates from weak signals rather than requiring ideal training distributions.

2.1 Quantum Error Mitigation

  • Analytic mitigation methods use assumed noise structure or known symmetries, while their sampling overhead grows exponentially with circuit depth.The reviewed approaches include zero-noise extrapolation, probabilistic error cancellation, global depolarizing corrections, and symmetry-based methods.Source: §2.1 · p. 2

2.2 Machine Learning Approaches to Error Mitigation

  • Learned mitigation either transfers corrections from simulable circuit families or trains on ideal-noisy pairs, which still require noiseless simulation.The latter includes flexible neural approaches, while the former assumes transfer from restricted, non-universal circuit families.Source: §2.2 · pp. 2–3
  • This work replaces restricted-family transfer or direct simulation with an aggregate of unreliable, inexpensive signals.It offers a third source of training signal for learned mitigation.Source: §2.2 · p. 3

2.3 Measurement and Readout Error Mitigation

  • The mitigation step uses tensor-product confusion-matrix inversion, but predicts rates per circuit rather than taking one device-wide rate from calibration.Holding the correction method fixed isolates the source of the rates as the difference from calibration-based methods.Source: §2.3 · p. 3

2.4 Weak Supervision and Programmatic Labeling

  • Weak supervision aggregates cheap, noisy proxies through labeling functions, and this paper applies it to quantum error mitigation using physical signals and per-qubit probabilities.Prior applications concentrated on text, images, and clinical records.Source: §2.4 · p. 3

2.5 Label Models and Learning from Noisy Labels

  • The label model uses the Dawid-Skene generative approach, while downstream learning treats its probabilities as noisy labels and calibrates their magnitude.The paper uses Platt scaling on a held-out slice because mitigation consumes rates, not just rankings.Source: §2.5 · p. 3

3 Problem Formulation

The paper formulates mitigation as predicting each qubit’s probability of measurement corruption and applying a map to move noisy output distributions closer to ideal ones. It restricts training targets to instances where the ideal outcome is determined and bars ideal-distribution information from influencing training labels.

3.1 Setting

  • For a circuit on n qubits, mitigation seeks a map that moves its empirical noisy distribution closer to the corresponding noiseless distribution.The circuit is executed for S shots and produces an empirical distribution over 2^n outcomes.Source: §3.1 · p. 4
  • The target is a calibrated probability that an individual qubit’s measured outcome was corrupted relative to its ideal outcome.The corruption indicator is defined per qubit and circuit.Source: §3.1 · p. 4

3.2 The Ideal-Label Bottleneck

  • No quantity derived from the ideal distribution may influence labels used to fit the model; ideal outputs are reserved for evaluation and a disjoint small calibration slice.For general circuits, producing the ideal distribution requires simulation with exponential cost in qubit count.Source: §3.2 · p. 4

3.3 A Well-Posed Target from Aggregated Distributions

  • With aggregated distributions, the target is formed by marginalizing ideal and measured distributions onto each qubit because per-shot bitstrings are unavailable.This provides a target from the released prior-work datasets without recovering individual shots.Source: §3.3 · p. 4
  • A qubit-instance with a determined ideal outcome treats deviation in the measured marginal as corruption.When the ideal bit is 0 or 1, any measured marginal deviation identifies corruption.Source: §3.3 · p. 4
  • Instances whose ideal marginal is not near 0 or 1 are excluded because measured shifts cannot be attributed to corruption; admissibility is fixed before fitting.An instance is admissible when min(m⋆_c,i, 1 − m⋆_c,i) ≤ τ.Source: §3.3 · p. 4

4 Method

The method turns votes from labeling functions into per-qubit error probabilities, generalizes those probabilities to unseen circuits, and interprets them as a readout channel for mitigation. The same channel inversion is used for learned and classical corrections so their rate sources can be compared.

4.1 Overview

  • Predicted soft labels are generalized from training circuits to unseen circuits, interpreted as readout error rates, and used to invert the measured distribution.Only final evaluation touches the ideal distribution.Source: §4.1 · p. 4, Figure 1 · p. 5

4.2 Labeling Functions

  • Each labeling function votes CLEAN, ERROR, or ABSTAIN on a circuit-qubit instance, with abstentions treated as missing votes rather than guesses.Sixteen functions use non-simulator quantities; the matrix drops uninformative columns before label aggregation.Source: §4.2 · pp. 4–5
  • Labeling functions use measured parity and instability, calibration telemetry, and circuit structure; one stability filter abstains on these datasets because no noise-scaling sweep is available.The table describes the sixteen functions and their coverage and firing rates on the Algiers test split.Source: §4.2 · p. 5, Table 1 · p. 6

4.3 Label Model

  • Majority vote, precision-weighted votes, and a Dawid-Skene model aggregate votes into per-instance soft labels, with zero-variance columns removed and probabilities recalibrated.Recalibration preserves ranking while correcting scale.Source: §4.3 · p. 6

4.4 Downstream Model

  • A discriminative model trained on tabular per-qubit features predicts soft-label probabilities for unseen circuits, using splits separated by circuit.Evaluated models include ridge regression, logistic regression, and random forest.Source: §4.4 · p. 6

4.5 From Error Rates to Mitigation

  • Predicted error probabilities define independent per-qubit bit-flip channels whose tensor-product inversion mitigates the measured distribution.Using the same inversion for learned and classical SPAM corrections makes their rate sources the comparison; the dataset’s admissible instances have ideal marginals near 0 or 1.Source: §4.5 · p. 6, Table 2 · p. 7

5 Experimental Setup

The experiments use five-qubit Pauli-circuit data from IBM Algiers and Hanoi, with evaluation on a held-out test split. They compare analytical and supervised baselines using divergence metrics and paired bootstrap evaluation.

5.1 Data

  • At τ = 0.01, only 7.46% of qubit-instances have a determined ideal outcome; Pauli circuits yield three to four times more admissible instances than random circuits at every tolerance.The dataset contains five-qubit circuits executed on Algiers and Hanoi, and uses its train, validation, and test division so evaluation circuits are unseen during training.Source: §5.1 · p. 7

5.2 Baselines

  • The comparison includes SPAM, Expander, their combination, doing nothing, and supervised models trained directly on measured targets.Expander estimates fidelity analytically from the calibration snapshot, avoiding additional device executions; the supervised comparator requires labels unavailable for scalable deployment.Source: §5.2 · p. 7

5.3 Metrics

  • Mitigation quality is measured by KL divergence, total variation, and Hellinger distance; relative KL changes and the fraction of circuits improved address limitations of mean absolute KL.Absolute KL is inflated by the numerical floor because ideal distributions contain exact zeros; methods are scored on the same circuits, so differences and relative changes remain meaningful.Source: §5.3 · p. 8

5.4 Protocol

  • All methods use the same test circuits, with 95% bootstrap confidence intervals over 10,000 resamples and three seeds per configuration.Paired comparisons report mean differences, intervals, win rates, and Wilcoxon signed-rank tests; the authors treat intervals and win rates as conveying magnitude and reliability.Source: §5.4 · p. 8

6 Results

Across two devices, weak supervision removes more KL divergence than the published analytical baselines, while fully supervised models remain stronger at ranking individual qubit errors. Results also show that performance depends on signal coverage, rate calibration, and whether readout error is the dominant noise source.

6.2 Statistical Comparison

  • Seed variation is negligible, and the weak-supervision advantage over SPAM + Expander lies far outside its paired-comparison interval on both devices.The advantage wins on roughly seven circuits in ten, with effect sizes reported separately by device because its magnitude varies.Source: §6.2 · p. 8

6.3 Per-Qubit Prediction

  • Full supervision is roughly 0.11 AUROC better at per-qubit corruption ranking, but weak supervision trains on all instances rather than 7.4%.The weak path uses 13.5 times as many training instances, and its broader coverage produces a better correction in aggregate despite less informative labels per instance.Source: §6.3 · pp. 8–9

6.1 Mitigation Quality

  • 24.3% (Algiers) and 28.8% (Hanoi) of KL divergence is removed by the label model, versus 15.4% and 21.5% for SPAM + Expander.The label model and its ridge variant rank first and second on four of five Table 3 columns on both devices; SPAM leads in the fraction of circuits improved.Source: §6.1 · p. 8, Table 3 · p. 9

6.5 Which Signals Carry the Information

  • Eight of sixteen labeling functions contribute; measurement- or circuit-based functions contribute, while most calibration-snapshot-only functions do not.The stability filter and calibration-drift check have no coverage because the released data lacks the required noise-scaling sweep and reference snapshot.Source: §6.5 · p. 10

6.4 Context: What Full Supervision Achieves

  • With features, downstream model, and mitigation fixed, weak labels win on ΔKL in all six comparisons against training on the measured target.Supervised ridge nevertheless reaches lower absolute KL on both devices while removing less than half as much divergence relative to no mitigation.Source: §6.4 · p. 10, Table 6 · p. 11

6.7 Transfer to Expectation-Value Benchmarks

  • On the benchmark with readout error removed, the label model is worse than doing nothing (0.0686 against 0.0647), while direct expectation-value regression lowers MAE from 0.0525 to 0.0187.The readout-channel correction is a limitation when readout error is not the dominant noise source.Source: §6.7 · pp. 11–12

6.6 Labeling-Function Thresholds

  • Fitting labeling-function thresholds raises contributing functions from 8 to 12 and improves ranking by 0.04–0.06 AUROC, but slightly worsens mitigation.The random-forest variant falls to −0.022; rate magnitude, not ranking alone, matters to the channel correction, so threshold fitting is disabled in the reported configuration.Source: §6.6 · p. 11, Table 7 · p. 12

7 Discussion

The discussion attributes the learned rates’ advantage over calibration to larger, circuit-dependent corruption, while clarifying the gap to fully supervised models and the downstream model’s limited added value. It also identifies benchmark scope and ideal-output use as limitations and lists unrun analyses and future directions.

7.1 Why the Learned Rates Outperform Calibration

  • Measured corruption averages 0.1191 versus device readout error 0.0104, about eleven times larger; learned rates also capture per-qubit, per-circuit variation using local circuit features.The expectation-value transfer benchmark improves on unmitigated values but falls short of the published per-qubit random forest; without readout error, the label model is worse than no mitigation.Source: §7.1 · p. 12, Table 8 · p. 13

7.2 What the Supervised Models Buy

  • Without ideal labels, the constrained model recovers roughly two thirds of the unconstrained supervised model’s result; correlated readout, coherent gate error, and other unrepresented structure remain outside its channel.The reported sequence-model gap is 0.11 to 0.15 in relative divergence, and the comparison differs in both training information and model degrees of freedom.Source: §7.2 · p. 13

7.3 The Downstream Model Adds Little

  • On both devices, the label model alone matches or slightly exceeds the discriminative model trained on its output, making labeling-function quality the binding constraint.The authors interpret this as evidence that the features add no information beyond what the labeling functions already supply.Source: §7.3 · p. 13

7.4 Limitations

  • The method uses a disjoint ideal-output validation slice to recalibrate rates, and the benchmark’s admissible set is also selected using ideal outputs.Removing the validation slice remains open; selecting the admissible set before fitting does not remove the benchmark-population dependence on ideal outputs.Source: §7.4 · p. 13
  • Evidence is limited to five-qubit Pauli circuits on two superconducting devices; the tensor-product channel assumes independent flips, and larger systems, other circuits, and other hardware remain untested.Absolute KL values are inflated by exact zeros in ideal distributions, so the authors rely on differences and relative changes and corroborate ordering with bounded metrics.Source: §7.4 · p. 13

7.5 Future Work

  • Individual contributions of the eight useful labeling functions remain untested, as do shuffled-label and backend-identity-only controls.The proposed leave-one-out ablation would test whether results rest on a few signals or their aggregate.Source: §7.5 · p. 14
  • Cross-device transfer is untested, and the method’s readout-channel correction can become a liability when readout error is not the dominant noise source.A zero-noise-extrapolation comparison is also absent because the released data lack noise-amplified executions.Source: §7.5 · p. 14
  • The authors propose exploring deeper use of measured distributions and whether post-threshold calibration can preserve ranking gains while restoring rate scale.They also suggest calibrating labeling functions on a large corpus of quantum circuits.Source: §7.5 · p. 14

8 Conclusion

Weak supervision substitutes for ideal labels at a modest cost, outperforming published analytical baselines on two IBM devices while remaining weaker than supervised sequence models where ideal labels are available. Its value is in the regime where exponentially costly ideal-label training is unsupported.

8 Conclusion

  • Weak supervision removes more divergence to the ideal distribution than published analytical baselines on two IBM devices, with an advantage outside its bootstrap interval.The method gives up some performance relative to supervised sequence models where ideal labels exist.Source: §8 · p. 14
  • Supervised sequence models remain stronger by 0.11 to 0.15 in relative divergence when ideal labels are available.Source: §8 · p. 14
  • The method’s claim is to settings without ideal labels, whose production cost grows exponentially with system size.The authors characterize the penalty for giving up ideal labels as small enough to justify when no trained mitigator is otherwise available.Source: §8 · p. 14

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