Quantum Error Correction and $Z(2)$ Lattice Gauge Theories

TL;DR

This paper explores the connection between $Z(2)$ lattice gauge theories and quantum error correction, using Monte Carlo simulations to analyze phase diagrams relevant for quantum code thresholds under realistic noise models.

Contribution

It introduces a new $Z(2)$ gauge theory model capturing key aspects of quantum error correction under noise, and discusses Monte Carlo simulation results for these models.

Findings

Monte Carlo simulations provide insights into phase transitions relevant for QEC thresholds.

The $Z(2)$ gauge models effectively represent the effects of realistic noise on quantum codes.

Preliminary results indicate potential for improved understanding of error correction limits.

Abstract

$Z(2)$ lattice gauge theory plays an important role in the study of the threshold probability of Quantum Error Correction (QEC) for a quantum code. For certain QEC codes, such as the well-known Kitaev's toric/surface code, one can find a mapping of the QEC decoding problem onto a statistical mechanics model for a given noise model. The investigation of the threshold probability then corresponds to that of the phase diagram of the mapped statistical mechanics model. This can be studied by Monte Carlo simulation of the statistical mechanics model. In~\cite{Rispler}, we investigate the effects of realistic noise models on the toric/surface code in two dimensions together with syndrome measurement noise and introduce the random coupled-plaquette gauge model, 3-dimensional $Z(2) \times Z(2)$ lattice gauge theory. This new Z(2) gauge theory model captures main aspects of toric/surface code under depolarizing and syndrome noise. In these proceedings, we mainly focus on the aspects of Mont Carlo simulation and discuss preliminary results from Monte Carlo simulations of mapped classes of Z(2) lattice theories.