Multivariate Multicycle Codes for Complete Single-Shot Decoding

TL;DR

This paper introduces multivariate multicycle (MM) quantum error-correcting codes that enable complete single-shot decoding with high rates and distances, unifying various code families and offering new explicit constructions.

Contribution

The paper presents a new family of quantum codes, MM codes, with a general framework for construction, enabling efficient decoding and potential for logical non-Clifford gates.

Findings

Identified several high-performance MM code candidates with specific parameters.

Demonstrated codes surpass all known single-shot decodable quantum CSS codes.

Provided explicit boundary and metacheck matrices for code construction.

Abstract

We introduce multivariate multicycle (MM) codes, a new family of quantum error correcting codes that unifies and generalizes bivariate bicycle codes, multivariate bicycle codes, abelian two-block group algebra codes, generalized bicycle codes, trivariate tricycle codes, and n-dimensional toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-t chain complexes with $t \ge 4$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding, while also having additional algebraic structure that might enable logical non-Clifford gates. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$, $[[96, 44, 4]]$ $[[144, 40, 4]]$, $[[216, 12, 12]]$, $[[360, 30, 6]]$, $[[384, 80, 4]]$, $[[486, 24, 12]]$, $[[486, 66, 9]]$ and $[[648, 60, 9]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot decodable quantum CSS codes of practical blocksize.