Constructions and Noise Threshold of Topological Subsystem Codes

TL;DR

This paper explores topological subsystem codes, linking them to Kitaev's honeycomb model, and demonstrates their error correction capabilities with a threshold of at least 2% against depolarizing noise.

Contribution

It introduces a systematic construction of topological subsystem codes via hypergraph generalizations and analyzes their error correction thresholds.

Findings

The five-squares code has a depolarizing noise threshold of at least 2%.

A necessary and sufficient condition for syndrome measurement reduction is derived.

Decoding algorithms for the five-squares code are proposed and tested.

Abstract

Topological subsystem codes proposed recently by Bombin are quantum error correcting codes defined on a two-dimensional grid of qubits that permit reliable quantum information storage with a constant error threshold. These codes require only the measurement of two-qubit nearest-neighbor operators for error correction. In this paper we demonstrate that topological subsystem codes (TSCs) can be viewed as generalizations of Kitaev's honeycomb model to trivalent hypergraphs. This new connection provides a systematic way of constructing TSCs and analyzing their properties. We also derive a necessary and sufficient condition under which a syndrome measurement in a subsystem code can be reduced to measurements of the gauge group generators. Furthermore, we propose and implement some candidate decoding algorithms for one particular TSC assuming perfect error correction. Our Monte Carlo simulations indicate that this code, which we call the five-squares code, has a threshold against depolarizing noise of at least 2%.