Efficient computation of topological order

TL;DR

This paper compares two methods for detecting topological order in quantum systems, showing that an error correction-based approach scales better and enables analysis of larger systems than traditional entanglement entropy methods.

Contribution

It introduces an operational error correction-based definition of topological order and demonstrates its computational advantages over entanglement entropy, especially for larger systems.

Findings

Error correction method scales polynomially with system size.

Topological phase diagram can be computed using error correction.

Entanglement entropy is limited by finite size effects.

Abstract

We analyze the computational aspects of detecting topological order in a quantum many-body system. We contrast the widely used topological entanglement entropy with a recently introduced operational definition for topological order based on error correction properties, finding exponential scaling with the system size for the former and polynomial scaling for the latter. We exemplify our approach for a variant of the paradigmatic toric code model with mobile particles, finding that the error correction method allows to treat substantially larger system sizes. In particular, the phase diagram of the model can be successfully computed using error correction, while the topological entanglement entropy is too severely limited by finite size effects to obtain conclusive results. While we mainly focus on one-dimensional systems whose ground states can be expressed in terms of matrix product states, our strategy can be readily generalized to higher dimensions and systems out of equilibrium, even allowing for an efficient detection of topological order in current quantum simulation experiments.