Almost-linear time decoding algorithm for topological codes

TL;DR

This paper introduces a highly efficient decoding algorithm for topological quantum codes, capable of correcting errors rapidly with near-linear complexity, and demonstrates its optimality and practical thresholds.

Contribution

The authors develop a decoding algorithm with near-linear worst-case complexity for topological codes, improving speed and efficiency over previous methods.

Findings

Algorithm has worst-case complexity of O(n α(n))

Achieves a threshold of 9.9% for perfect measurements

Achieves a threshold of 2.6% with faulty measurements

Abstract

In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of $O(n \alpha(n))$, where $n$ is the number of physical qubits and $\alpha$ is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, $\alpha(n) \leq 3$. We prove that our algorithm performs optimally for errors of weight up to $(d-1)/2$ and for loss of up to $d-1$ qubits, where $d$ is the minimum distance of the code. Numerically, we obtain a threshold of $9.9\%$ for the 2d-toric code with perfect syndrome measurements and $2.6\%$ with faulty measurements.