Gauge Structures: From Stabilizer Codes to Continuum Models

TL;DR

This paper introduces a gauge structure framework for stabilizer codes, linking discrete quantum error-correcting codes to continuum gauge theories, and applies it to topological and fracton models to understand their properties.

Contribution

It formalizes stabilizer codes as a linear gauge structure, connecting them to continuum gauge theories and enabling new insights into their error correction and topological features.

Findings

Stabilizer codes can be viewed as a discrete gauge theory.

The formalism applies to toric codes and fracton models.

Continuum models capture key features of discrete codes.

Abstract

Stabilizer codes are a powerful method for implementing fault-tolerant quantum memory and in the case of topological codes, they form useful models for topological phases of matter. In this paper, we discuss the theory of stabilizer codes as a discrete version of a linear gauge structure, a concept we introduce here. A linear gauge structure captures all the familiar ingredients of $U(1)$ gauge theory including a generalization of charge conservation, Maxwell equations and topological terms. Using this connection, we prove some important results for stabilizer codes which can be used to characterize their error-correction properties. However, this perspective does not depend on any particular Hamiltonian or Lagrangian, which as a consequence is agnostic to the source of the gauge redundancy. Based upon the connection to stabilizer codes, we view the source of the gauge redundancy as an ambiguity in the tensor product structure of the Hilbert space. That is, the gauge provides a set of equivalent but arbitrary ways to factorize the Hilbert space. We apply this formalism to the $d=2$ and $3$ toric code as well as the paradigm fracton models, X-Cube and Haah's cubic code. From this perspective, we are able to map all these models to some continuum version which captures all or some of the fractonic and topological features of their parent discrete models.