Topological error correcting processes from fixed-point path integrals

TL;DR

This paper introduces a unified framework linking topological quantum error correction codes to fixed-point path integrals, revealing new codes and unifying existing ones through a geometric and topological perspective.

Contribution

It develops a formalism connecting topological codes with path integrals, unifies several codes under this framework, and derives new error-correcting codes with novel properties.

Findings

Unified description of topological codes via fixed-point path integrals

Identification of equivalences among known codes like toric and Floquet codes

Construction of new codes, including a 2-body measurement 3+1D toric code and a double-semion string-net code

Abstract

We propose a unifying paradigm for analyzing and constructing topological quantum error correcting codes as dynamical circuits of geometrically local channels and measurements. To this end, we relate such circuits to discrete fixed-point path integrals in Euclidean spacetime, which describe the underlying topological order: If we fix a history of measurement outcomes, we obtain a fixed-point path integral carrying a pattern of topological defects. As an example, we show that the stabilizer toric code, subsystem toric code, and CSS Floquet code can be viewed as one and the same code on different spacetime lattices, and the honeycomb Floquet code is equivalent to the CSS Floquet code under a change of basis. We also use our formalism to derive two new error-correcting codes, namely a Floquet version of the $3+1$-dimensional toric code using only 2-body measurements, as well as a dynamic code based on the double-semion string-net path integral.