Subsystem codes with spatially local generators

TL;DR

This paper introduces a family of 2D subsystem codes with local generators that achieve proportional logical qubits and distance, and demonstrates their potential advantages over stabilizer codes under spatial locality constraints.

Contribution

The paper constructs a new family of 2D subsystem codes with local generators and proportional parameters, and establishes bounds showing their superior potential compared to stabilizer codes.

Findings

Existence of 2D subsystem codes with proportional k and d

Subsystem codes can outperform stabilizer codes under locality constraints

Introduction of generalized Bacon-Shor codes with interesting properties

Abstract

We study subsystem codes whose gauge group has local generators in the 2D geometry. It is shown that there exists a family of such codes defined on lattices of size LxL with the number of logical qubits k and the minimum distance d both proportional to L. The gauge group of these codes involves only two-qubit generators of type XX and ZZ coupling nearest neighbor qubits (and some auxiliary one-qubit generators). Our proof is not constructive as it relies on a certain version of the Gilbert-Varshamov bound for classical codes. Along the way we introduce and study properties of generalized Bacon-Shor codes which might be of independent interest. Secondly, we prove that any 2D subsystem [n,k,d] code with spatially local generators obeys upper bounds kd=O(n) and d^2=O(n). The analogous upper bound proved recently for 2D stabilizer codes is kd^2=O(n). Our results thus demonstrate that subsystem codes can be more powerful than stabilizer codes under the spatial locality constraint.