Calderbank-Shor-Steane codes on group-valued qudits

TL;DR

This paper introduces a new class of quantum error-correcting codes on group-valued qudits, generalizing CSS codes and Kitaev quantum doubles, with applications to non-Abelian groups and topological states.

Contribution

It extends CSS codes to group-valued qudits, generalizes quantum double models to CW complexes, and explores non-Abelian codes with optimal parameters and boundary conditions.

Findings

Codes on non-Abelian simple groups have provable distance bounds.

Constructed non-Abelian codes with asymptotically optimal rate and distance.

Generalized quantum double models with defects and boundary conditions.

Abstract

Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting $X$- and $Z$-type checks. We introduce CSS-like codes on $G$-valued qudits for any finite group $G$ that reduce to qubit CSS codes for $G = \mathbb{Z}_2$ yet generalize the Kitaev quantum double model for general groups. The $X$-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while $Z$-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when $G$ is non-Abelian and simple, every $G$-covariant group-CSS code with suitably upper-bounded $Z$-check weight and lower-bounded $Z$-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.