When Noisy Quantum Order Finding Remains Recoverable for Shor's Algorithm

2026-05-15

AI Summary

In noisy environments, the output distribution of Shor’s quantum order-finding subroutine is prone to distortion, hindering accurate recovery of the true order. Based on 680 sets of phase estimation data from IBM quantum devices, this work systematically characterizes the key features governing recoverability in noisy order finding by integrating continued fraction expansion, modular validation, autocorrelation analysis, normalized entropy, and tree-based machine learning. The study reveals that the dominant validation quality score serves as the strongest single-variable predictor of success. Notably, some highly distorted distributions remain accurately recoverable when a single candidate denominator dominates, whereas certain structurally clear distributions fail due to post-processing misclassification. Interpretable decision tree rules derived from the analysis uncover the underlying mechanisms determining recovery outcomes.

Abstract

Order finding is the core subroutine of Shor's algorithm. On NISQ hardware, phase estimation output distributions are often distorted by noise, making correct order recovery difficult. We study recoverability in noisy order finding: given a measured precision-register distribution, when does standard classical post-processing still return the true order? We analyze 680 distributions from IBM quantum systems across problem instances and circuit settings. For each distribution, we apply continued-fraction post-processing with modular verification and define recoverability as whether the recovered order equals the true one. We characterize each distribution using four features: autocorrelation peak strength, normalized entropy, dominant verified mass fraction, and verified margin fraction. We evaluate these quantities using marginal feature comparisons, single-feature AUROC analysis, and multivariate tree-based classifiers. We use random-forest permutation importance to assess which quantities contribute distinct predictive information once the other features are known. To make classification behavior interpretable, we train a decision tree that exposes threshold rules for recoverable and non-recoverable distributions. We find that recoverability is strongly associated with residual comb-like structure in the measured distribution and the way verified probability mass is organized across candidate denominators. The dominant verified mass fraction is the strongest single-feature indicator of recoverability, and tree-based analysis shows it also provides the primary split in an interpretable threshold description. Some highly distorted distributions remain recoverable when one verified denominator dominates the post-processing mass, while some visibly structured distributions fail because classical post-processing favors an incorrect verified denominator.

Problem

• noisy quantum order finding
• Shor's algorithm
• phase estimation
• order recovery
• NISQ

Innovation

• noisy order finding
• Shor's algorithm
• classical post-processing
• recoverability
• interpretable decision tree