Intrinsic Error Thresholds in Nearly Critical Toric Codes

TL;DR

This paper investigates the robustness of nearly critical topological quantum codes, specifically the transverse-field toric code, against local decoherence, revealing a finite error threshold due to the irrelevance of certain defects at criticality.

Contribution

It introduces a novel analysis of error thresholds in nearly critical topological codes using a replica statistical physics approach and field theory, demonstrating the irrelevance of defects at the critical point.

Findings

Finite decoherence strength is needed to irreversibly destroy information.

Defects introduced by decoherence are perturbatively irrelevant at the critical point.

The results suggest similar thresholds for a broad class of nearly critical topological codes.

Abstract

We study the protection of information in nearly critical topological quantum codes, constructed by perturbing topological stabilizer codes towards continuous quantum phase transitions. Our focus is on the transverse-field toric code subjected to local Pauli decoherence. Despite the strong quantum fluctuations of anyons when the transverse field is tuned infinitesimally close to the critical point, we show that a finite strength of Pauli decoherence remains necessary to irreversibly destroy information encoded in the ground-state manifold. Using a replica statistical physics mapping for the coherent information, we show that decoherence can be understood as introducing a two-dimensional inter-replica defect within a three-dimensional replica statistical physics model. A field theoretical analysis shows that this defect is perturbatively irrelevant to the bulk critical point, and cannot renormalize the transverse field strength, leading to a finite error threshold. We argue that a qualitatively similar conclusion can be drawn for a broad class of nearly critical topological codes, under a variety of decoherence channels.