Gauss law codes and vacuum codes from lattice gauge theories

TL;DR

This paper introduces a unified framework for constructing quantum error correcting codes from Abelian lattice gauge theories, leveraging quantum reference frames to analyze their structure and error correction capabilities.

Contribution

It develops a comprehensive formalism for QECCs from Abelian LGTs, including gauge-invariant and vacuum codes, with systematic characterization using QRFs across various gauge groups.

Findings

Vacuum codes are unitarily equivalent to Gauss law codes for finite groups.

QRFs systematically characterize code algebraic structures and error sets.

Illustrations include $ ext{Z}_2$-gauge theory, scalar, and fermionic QED.

Abstract

We develop a comprehensive framework for constructing quantum error correcting codes (QECCs) from Abelian lattice gauge theories (LGTs) using quantum reference frames (QRFs) as a unifying formalism. We consider LGTs with arbitrary compact Abelian gauge groups supported on lattices in arbitrary numbers of spatial dimensions, and we work with both pure gauge theories and theories with couplings to bosonic and fermionic matter. The codes that we construct fall into two classes: First, Gauss law codes identify the code subspace with the full gauge-invariant sector of the theory. In models with matter coupled to gauge fields, these codes inherit a natural subsystem structure in which gauge-invariant Wilson loops and dressed matter excitations factorize the code space. Second, vacuum codes restrict the code subspace to the matter vacuum sector within the gauge-invariant subspace, yielding codes where errors correspond to gauge-invariant charge excitations rather than to violations of the Gauss law. Despite their distinct setup, we show that when the gauge group is finite, vacuum codes are unitarily equivalent to pure gauge theory Gauss law codes, and that when the group is continuous, this is only true upon a charge coarse-graining of the vacuum code. In all cases, QRFs provide a systematic apparatus for fully characterizing the codes' algebraic structures and correctable error sets. For clarity, we illustrate our general results in $\mathbb{Z}_2$-gauge theory, as well as in scalar and fermionic QED. These findings offer fundamental insights into the parallelism between quantum error correction and gauge theory and point toward practical advantages for simulating LGTs on noisy quantum devices.