Fail fast: techniques to probe rare events in quantum error correction

TL;DR

This paper introduces three innovative techniques to efficiently analyze rare logical failure events in quantum error correction codes, enabling better performance assessment of high-fidelity logical qubits.

Contribution

It develops a low-parameter failure spectrum ansatz, computes min-weight failing configurations, and generalizes the splitting method for qLDPC codes, advancing rare-event analysis in quantum error correction.

Findings

Effective failure spectrum prediction across error rates

Exact enumeration of min-weight failing configurations

Enhanced convergence in rare-event sampling for complex codes

Abstract

The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.