Self-dual random-plaquette gauge model and the quantum toric code

TL;DR

This paper analyzes a four-dimensional Z_2 lattice gauge theory related to the quantum toric code, revealing its phase structure and accuracy threshold for quantum error correction using duality and replica methods.

Contribution

It establishes a precise mathematical connection between the 4D random-plaquette gauge model and the 2D random-bond Ising model, enabling exact predictions of the error correction threshold.

Findings

Exact multicritical point p_c=0.889972... for the model

Demonstrates the phase boundary determines error correction accuracy

Provides bounds on the threshold in three dimensions

Abstract

We study the four-dimensional Z_2 random-plaquette lattice gauge theory as a model of topological quantum memory, the toric code in particular. In this model, the procedure of quantum error correction works properly in the ordered (Higgs) phase, and phase boundary between the ordered (Higgs) and disordered (confinement) phases gives the accuracy threshold of error correction. Using self-duality of the model in conjunction with the replica method, we show that this model has exactly the same mathematical structure as that of the two dimensional random-bond Ising model, which has been studied very extensively. This observation enables us to derive a conjecture on the exact location of the multicritical point (accuracy threshold) of the model, p_c=0.889972..., and leads to several nontrivial results including bounds on the accuracy threshold in three dimensions.