Ungauging quantum error-correcting codes

TL;DR

This paper introduces a systematic framework for gauging and ungauging in quantum error-correcting codes, revealing new models with unusual symmetries and connections to topological phases, including a fracton SPT phase.

Contribution

It develops a formalism for gauging and ungauging in quantum codes, demonstrating applications to various models and constructing new topological phases with complex symmetries.

Findings

Ungauging can produce models with lower-dimensional symmetry operators.

The codeword space of the 3D gauge color code relates to multiple copies of lattice gauge theory.

Explicit realization of a fracton SPT phase with fractal symmetries.

Abstract

We develop the procedures of gauging and ungauging, reveal their operational meaning and propose their generalization in a systematic manner within the framework of quantum error-correcting codes. We demonstrate with an example of the subsystem Bacon-Shor code that the ungauging procedure can result in models with unusual symmetry operators constrained to live on lower-dimensional structures. We apply our formalism to the three-dimensional gauge color code (GCC) and show that its codeword space is equivalent to the Hilbert space of six copies of $\mathbb{Z}_2$ lattice gauge theory with $1$-form symmetries. We find that three different stabilizer Hamiltonians associated with the GCC correspond to distinct thermal symmetry-protected topological (SPT) phases in the presence of the stabilizer symmetries of the GCC. One of the considered Hamiltonians describes the Raussendorf-Bravyi-Harrington model, which is universal for measurement-based quantum computation at non-zero temperature. We also propose a general procedure of creating $D$-dimensional SPT Hamiltonians from $(D+1)$-dimensional CSS stabilizer Hamiltonians by exploiting a relation between gapped domain walls and transversal logical gates. As a result, we find an explicit two-dimensional realization of a non-trivial fracton SPT phase protected by fractal-like symmetries.