Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models

TL;DR

This paper introduces a systematic way to construct three-dimensional topological subsystem codes from two-dimensional anyon models, enhancing quantum error correction with potential single-shot capabilities and exploring their physical properties.

Contribution

It generalizes the subsystem toric code to a broader class of 3D codes derived from 2D models, providing new insights into their structure and physical realization.

Findings

Codes exhibit single-shot error correction properties

Numerical simulations confirm robustness against phenomenological noise

Associated Hamiltonians may be gapless

Abstract

Topological subsystem codes in three spatial dimensions allow for quantum error correction with no time overhead, even in the presence of measurement noise. The physical origins of this single-shot property remain elusive, in part due to the scarcity of known models. To address this challenge, we provide a systematic construction of a class of topological subsystem codes in three dimensions built from abelian quantum double models in one fewer dimension. Our construction not only generalizes the recently introduced subsystem toric code [Kubica and Vasmer, Nat. Commun. 13, 6272 (2022)] but also provides a new perspective on several aspects of the original model, including the origin of the Gauss law for gauge flux, and boundary conditions for the code family. We then numerically study the performance of the first few codes in this class against phenomenological noise to verify their single-shot property. Lastly, we discuss Hamiltonians naturally associated with these codes, and argue that they may be gapless.