CSS code surgery as a universal construction

TL;DR

This paper introduces a universal framework for CSS code surgery using chain complex maps, enabling fault-tolerant merging and splitting of codes while preserving LDPC properties.

Contribution

It presents a novel categorical approach to CSS code surgery via chain complex colimits, providing a general recipe for creating new codes and fault-tolerant operations.

Findings

Defines code maps between CSS codes using chain complexes.

Describes code surgery as a colimit in the category of chain complexes.

Shows merging and splitting preserve LDPC properties.

Abstract

We define code maps between Calderbank-Shor-Steane (CSS) codes using maps between chain complexes, and describe code surgery between such codes using a specific colimit in the category of chain complexes. As well as describing a surgery operation, this gives a general recipe for new codes. As an application we describe how to `merge' and `split' along a shared $\overline{X}$ or $\overline{Z}$ operator between arbitrary CSS codes in a fault-tolerant manner, so long as certain technical conditions concerning gauge fixing and code distance are satisfied. We prove that such merges and splits on LDPC codes yield codes which are themselves LDPC.