Comparison of 2D topological codes and their decoding performances

TL;DR

This paper compares the decoding performance of various 2D topological quantum codes, including CSS and non-CSS types, using an adapted belief propagation method, highlighting their efficiency and robustness under perfect syndrome conditions.

Contribution

It demonstrates that diverse 2D topological codes can be decoded effectively with MBP, regardless of layout or code type, expanding decoding applicability.

Findings

MBP decoding applies to CSS and non-CSS codes

Color and twisted XZZX codes are included in the comparison

Decoding performance and code efficiency are comprehensively analyzed

Abstract

Topological quantum codes are favored because they allow qubit layouts that are suitable for practical implementation. An $N$-qubit topological code can be decoded by minimum-weight perfect matching (MWPM) with complexity $O(\text{poly}(N))$ if it is of CSS-type. Recently it is shown that various quantum codes, including non-CSS codes, can be decoded by an adapted belief propagation with memory effects (denoted MBP) with complexity almost linear in $N$. In this paper, we show that various two-dimensional topological codes, CSS or non-CSS, regardless of the layout, can be decoded by MBP, including color codes and twisted XZZX codes. We will comprehensively compare these codes in terms of code efficiency and decoding performance, assuming perfect error syndromes.