Computing logical error thresholds with the Pauli Frame Sparse Representation

TL;DR

This paper presents a new classical representation and sampling method to improve quantum error correction threshold predictions, revealing significant overestimations by traditional approximations and providing refined error rate bounds.

Contribution

The authors introduce a sparse classical representation and sampling strategy that enhance the accuracy of quantum error correction threshold predictions beyond existing Clifford and Pauli error models.

Findings

Coherent noise thresholds are overestimated by a factor of about 4 using Pauli-twirling approximation.

At distance d=5, the T-gate error rate can be up to 7 times larger than the S-gate error rate.

The method improves prediction accuracy for quantum error correction thresholds and error rates.

Abstract

We introduce a sparse classical representation, a truncation strategy and a shot-efficient sampling method to push the classical prediction of quantum error correction thresholds beyond Clifford operations and Pauli errors. As two illustrations of the potential of our method, we first show that coherent noise error thresholds, when computed at the circuit level (i.e taking into account full syndrome circuits) for distances up to d=9, are systematically overestimated (by a factor of about 4) by a Pauli-twirling approximation of the noise. We then apply our method to the recently introduced magic-state cultivation protocol. We show, through shot-efficient importance sampling, that, at distance d=5, the multiplicative factor between the T-gate and the S-gate injection error rate is not the one conjectured from low-d computations: it can be as large as 7.