Topological Subsystem Codes From Graphs and Hypergraphs

TL;DR

This paper introduces new families of topological subsystem codes derived from graphs and hypergraphs, offering novel constructions and efficient error recovery schemes that enhance quantum error correction capabilities.

Contribution

It presents previously unknown topological subsystem codes from hypergraphs and develops syndrome measurement schedules leveraging 2-local gauge groups.

Findings

New topological subsystem codes from hypergraphs

Efficient syndrome measurement schedules for error recovery

A general construction for color codes

Abstract

Topological subsystem codes were proposed by Bombin based on 3-face-colorable cubic graphs. Suchara, Bravyi and Terhal generalized this construction and proposed a method to construct topological subsystem codes using 3-valent hypergraphs that satisfy certain constraints. Finding such hypergraphs and computing their parameters however is a nontrivial task. We propose families of topological subsystem codes that were previously not known. In particular, our constructions give codes which cannot be derived from Bombin's construction. We also study the error recovery schemes for the proposed subsystem codes and give detailed schedules for the syndrome measurement that take advantage of the 2-locality of the gauge group. The study also leads to a new and general construction for color codes.