An efficient Markov chain Monte Carlo algorithm for the surface code

TL;DR

This paper introduces a new Markov chain Monte Carlo algorithm for surface code error correction that achieves lower logical error rates than traditional methods, with adjustable trade-offs between runtime and accuracy.

Contribution

The authors develop an MCMC algorithm that outperforms MWPM in logical error rates while maintaining polynomial runtime complexity, and it can be parallelized and handle measurement imperfections.

Findings

Lower logical error rates than MWPM for similar or reduced runtime.

Achieves exponentially smaller logical error rates as code size increases.

Allows trade-offs between runtime and error correction performance.

Abstract

Minimum-weight perfect matching (MWPM) has been been the primary classical algorithm for error correction in the surface code, since it is of low runtime complexity and achieves relatively low logical error rates [Phys. Rev. Lett. 108, 180501 (2012)]. A Markov chain Monte Carlo (MCMC) algorithm [Phys. Rev. Lett. 109, 160503 (2012)] is able to achieve lower logical error rates and higher thresholds than MWPM, but requires a classical runtime complexity which is super-polynomial in L, the linear size of the code. In this work we present an MCMC algorithm that achieves significantly lower logical error rates than MWPM at the cost of a polynomially increased classical runtime complexity. For error rates p close to the threshold, our algorithm needs a runtime complexity which is increased by O(L^2) relative to MWPM in order to achieve a lower logical error rate. If p is below an L-dependent critical value, no increase in the runtime complexity is necessary any longer. For p->0, the logical error rate achieved by our algorithm is exponentially smaller (in L) than that of MWPM, without requiring an increased runtime complexity. Our algorithm allows for trade-offs between runtime and achieved logical error rates as well as for parallelization, and can be also used to correct in the case of imperfect stabilizer measurements.