Three-dimensional color code thresholds via statistical-mechanical mapping

TL;DR

This paper estimates error thresholds for 3D color codes in quantum computing by mapping the problem to statistical mechanics models and using Monte Carlo simulations, revealing high robustness against certain errors.

Contribution

It introduces a novel connection between 3D color code error correction and statistical-mechanical models, providing threshold estimates via Monte Carlo methods.

Findings

Threshold for 1D string-like logical operators: ~1.9%.

Threshold for 2D sheet-like logical operators: ~27.6%.

Development of new 3D statistical-mechanical models for analysis.

Abstract

Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage threshold error rates for bit-flip and phase-flip noise in the 3D color code on the body-centererd cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we estimate the threshold for 1D string-like and 2D sheet-like logical operators to be $p^{(1)}_\mathrm{3DCC} \simeq 1.9\%$ and $p^{(2)}_\mathrm{3DCC} \simeq 27.6\%$. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the 4- and 6-body random coupling Ising models.