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Budget-Aware Neural Error Mitigation forVariational Quantum Algorithms

Guilin Zhang, Kai Zhao, Xu Chu

Published Aug 14, 2026• 0 citations
Publisher

Abstract

Abstract Variational quantum algorithms (VQAs) are limited by device noise, and error mitigation is the standard remedy on near-term hardware. Choosing a mitigation method for deployment requires comparing its true cost, yet the usual comparison hides it: it fixes the number of shots per circuit execution, while the methods differ sharply in how many executions they need per estimate. Zero-noise extrapolation (ZNE) needs three to five, Clifford data regression (CDR) more than thirty per target circuit, and a trained neural model exactly one; ZNE's accuracy further rests on a noise-scaling assumption that real devices violate. Inside a variational loop that evaluates thousands of parameter settings, it is this execution multiplicity, not the accuracy at unlimited shots, that decides which method is affordable. Our premise is that the comparison that matters is at equal total shot budget per evaluation point. We present budget-conditioned neural error mitigation (NEM): one network corrects a single shot-limited measurement from circuit and device features, conditioned on the shot-noise scale so that one model serves every budget. We evaluate every method—including each ZNE fold point and each CDR training circuit—under a protocol that spends the same total shots per evaluation point. Under it, NEM is the most accurate method at every budget under coherent miscalibration (reducing error by 72–81%), and the only method that beats unmitigated execution at budgets up to 2^14 shots under stochastic device noise (11–20%); under miscalibration CDR is also positive but overtakes NEM only at 2^16 shots, at 32x the per-evaluation cost. We further characterize when extrapolation fails: under control-offset miscalibration, folding ZNE returns a worse answer than no mitigation on 38–63% of instances in simulation, and on up to 88% on a real IBM Heron processor. On that processor, fine-tuning the model on a small on-device set—120 circuits with classically computed labels—narrows the simulation-to-hardware gap, recovering a 39% error reduction on held-out device instances from the same calibration epoch. Improvements hold from 4 to 20 qubits. Budget-aware accounting, we argue, is the comparison that matters for deployment.

DOI
10.21203/rs.3.rs-10093362/v1

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Abstract Variational quantum algorithms (VQAs) are limited by device noise, and error mitigation is the standard remedy on near-term hardware. Choosing a mitigation method for deployment requires comparing its true cost, yet the usual comparison hides it: it fixes the number of shots per circuit execution, while the methods differ sharply in how many executions they need per estimate. Zero-noise extrapolation (ZNE) needs three to five, Clifford data regression (CDR) more than thirty per target circuit, and a trained neural model exactly one; ZNE's accuracy further rests on a noise-scaling assumption that real devices violate. Inside a variational loop that evaluates thousands of parameter settings, it is this execution multiplicity, not the accuracy at unlimited shots, that decides which method is affordable. Our premise is that the comparison that matters is at equal total shot budget per evaluation point. We present budget-conditioned neural error mitigation (NEM): one network corrects a single shot-limited measurement from circuit and device features, conditioned on the shot-noise scale so that one model serves every budget. We evaluate every method—including each ZNE fold point and each CDR training circuit—under a protocol that spends the same total shots per evaluation point. Under it, NEM is the most accurate method at every budget under coherent miscalibration (reducing error by 72–81%), and the only method that beats unmitigated execution at budgets up to 2^14 shots under stochastic device noise (11–20%); under miscalibration CDR is also positive but overtakes NEM only at 2^16 shots, at 32x the per-evaluation cost. We further characterize when extrapolation fails: under control-offset miscalibration, folding ZNE returns a worse answer than no mitigation on 38–63% of instances in simulation, and on up to 88% on a real IBM Heron processor. On that processor, fine-tuning the model on a small on-device set—120 circuits with classically computed labels—narrows the simulation-to-hardware gap, recovering a 39% error reduction on held-out device instances from the same calibration epoch. Improvements hold from 4 to 20 qubits. Budget-aware accounting, we argue, is the comparison that matters for deployment.
Error mitigation is expected to turn noisy quantum measurements into more useful estimates. We show that it can instead amplify error. Under control-offset miscalibration, the evaluated unconstrained zero-noise extrapolation (ZNE) estimators are worse than no mitigation on 38-63% of simulated instances and produce extreme tail errors. A small IBM Heron study finds worsening on 80-88% of 18 instances across two depths, with mean error 3.0-4.1 times the raw error. In simulation, the median changes little, so median-only monitoring misses the failures. We then compare methods at the same online shot budget per evaluation and report offline training cost separately. After that cost is amortized, a budget-conditioned neural corrector occupies the low-budget end of the simulated accuracy-cost frontier and falls back to the raw estimate when an ensemble disagrees; its hardware transfer remains unsuccessful. Finally, we treat provider-reported calibration as an input that is not bound to the execution-time device state. A white-box projected-gradient stress test on this metadata increases the corrector's error 8.1 times within the feature ranges used for training and evades the disagreement monitor on half of the worsened cases. Together, the results connect budget-aware evaluation, tail risk, and calibration integrity in near-term quantum learning pipelines.
Achieving near-term quantum advantage will require effective methods for mitigating hardware noise. Data-driven approaches to error mitigation are promising, with popular examples including zero-noise extrapolation (ZNE) and Clifford data regression (CDR). Here, we propose a scalable error mitigation method that conceptually unifies ZNE and CDR. Our approach, called variable-noise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via near-Clifford circuits (which are classically simulable) and second by varying the noise levels in these circuits. We employ a noise model obtained from IBM's Ourense quantum computer to benchmark our method. For the problem of estimating the energy of an 8-qubit Ising model system, vnCDR improves the absolute energy error by a factor of 33 over the unmitigated results and by factors of 20 and 1.8 over ZNE and CDR, respectively. For the problem of correcting observables from random quantum circuits with 64 qubits, vnCDR improves the error by factors of 2.7 and 1.5 over ZNE and CDR, respectively.
We experimentally study error mitigation and finite-shot sampling on superconducting quantum hardware under a constrained execution budget. The study combines calibration-aware qubit selection, circuit-depth scaling, zero-noise extrapolation, dynamical decoupling, readout-error mitigation, and repeated-shot estimation on an IBM Quantum processor. Experiments are organized across ideal simulation, noise-model simulation, and physical-device execution to separate sampling uncertainty from device-induced error. We investigate how mitigation performance changes with circuit depth, effective noise scale, qubit connectivity, and measurement budget, and quantify accuracy using expectation-value error, mean-squared error, statistical uncertainty, and mitigation gain. A fixed hardware-execution budget is used to evaluate shot allocation strategies and repeated measurements without relying on unlimited sampling. The resulting analysis provides a hardware-aware characterization of when mitigation improves expectation-value estimation and when finite-shot fluctuations offset the benefit of additional mitigation overhead. The implementation uses contemporary Qiskit and IBM Quantum Runtime workflows and is designed to provide reproducible experimental evidence for error-mitigation studies on current quantum processors.
Quantum error mitigation (QEM) bridges the gap between today's noisy NISQ hardware and fault-tolerant quantum computing by reducing the effect of hardware errors on expectation value estimates without the qubit overhead of full quantum error correction. This paper proposes the Quantum Error Mitigation Design (QEMD) framework, a systematic evaluation of six QEM techniques across four circuit families on IBM Quantum Eagle and IonQ Forte: Zero-Noise Extrapolation (ZNE), Probabilistic Error Cancellation (PEC), Clifford Data Regression (CDR), Symmetry Verification (SV), Dynamical Decoupling (DD), and a novel Adaptive Ensemble Mitigation (AEM) combining ZNE and CDR with context-aware technique selection. QEMD finds: ZNE 3-point Richardson provides the best cost-performance trade-off for QAOA and VQE circuits (4.8x overhead, 28.4pp error reduction); PEC provides the highest accuracy but at 48x shot overhead -- practical only for n <= 12; DD reduces coherence-limited errors by 18.4% on IonQ Forte at zero shot overhead; AEM outperforms all individual techniques by 6.4pp at 8.4x overhead. QEMD provides a decision guide mapping circuit type, qubit count, and shot budget to optimal QEM technique selection.
Variational quantum circuits (VQCs) are central to near-term quantum computing, yet their practical deployment is severely hindered by noise. While existing error mitigation methods, such as zero-noise extrapolation (ZNE), typically assume static noise, real noisy intermediate-scale quantum (NISQ) systems exhibit dynamic, time-varying noise that remains largely unaddressed. To overcome this critical gap, our work introduces a novel adaptive noise mitigation framework for VQCs that integrates ZNE with contextual multi-armed bandits (CMAB), enabling dynamic, context-aware selection of circuit-folding levels based on ansatz parameters (e.g., depth, parameter count) and the evolving noise environment. Unlike fixed-fold or heuristic ZNE, our approach uses online adaptation to improve the accuracy of ZNE and reduce redundant quantum circuit executions. Our extensive simulations and experiments on real quantum hardware reveal the following important properties: (i) deeper VQCs accumulate noise, degrading accuracy and increasing the number of quantum circuit executions; (ii) ZNE restores estimator fidelity when the folding level is chosen appropriately; and (iii) CMAB-guided folding cuts quantum circuit execution round trips by up to 40\%, bytes exchanged by up to 35\%, and end-to-end cost by up to 30\% under a 10~Mbps budget, with up to 6.9\% higher estimator fidelity (CIFAR-10, depth 3, noise band η=0.05η=0.05η=0.05), versus fixed-fold and grid-search ZNE. These results demonstrate substantial performance gains over existing noise mitigation methods, underscoring the effectiveness of our design in supporting robust noise mitigation for VQCs. The source code is also publicly released to support reproducibility.
Machine Learning-based quantum error mitigation (ML-QEM) has emerged as a promising approach for improving the performance of noisy quantum algorithms. However, existing ML-QEM methods often have restricted applicability to variational circuits and rely on inaccessible noiseless training data. In this work, we propose a practical ML-QEM protocol tailored to variational quantum algorithms, which generates training data by simulating (near-)Clifford circuits. This data is used for model selection and training, producing a mitigation model that can correct variational circuits with arbitrary parameters and transfer across different target Hamiltonians of similar structure. We benchmark the proposed method on the Variational Quantum Eigensolver (VQE) task for the Sherrington-Kirkpatrick Hamiltonian of up to n=12n=12n=12 qubits under various noise models, analyzing its effect on trainability and comparing its performance against standard Zero-Noise Extrapolation (ZNE). The results demonstrate consistent several-fold error suppression across all tested settings and superior performance over ZNE in the high-noise regime, providing evidence for the applicability of the proposed protocol to present-day NISQ processors.