Phenomenological Noise Models and Optimal Thresholds of the 3D Toric Code

TL;DR

This paper analyzes the fault-tolerance thresholds of the 3D toric code under realistic noise, including measurement errors, and derives models to understand its robustness and practical performance in quantum computing.

Contribution

It introduces a novel analysis of the 3D toric code's thresholds with measurement errors using coupled lattice gauge models and duality techniques, providing new insights into its robustness.

Findings

Thresholds of approximately 11% for bit-flip errors.

Thresholds of approximately 2% for phase-flip errors.

Robustness of the 3D toric code against measurement errors.

Abstract

Three-dimensional (3D) topological codes offer the advantage of supporting fault-tolerant implementations of non-Clifford gates, yet their performance against realistic noise remains largely unexplored. In this work, we focus on the paradigmatic 3D toric code and investigate its fault-tolerance thresholds in the presence of both Pauli and measurement errors. Two randomly coupled lattice gauge models that describe the code's correctability are derived, including a random 2-form $\mathbb{Z}_2$ gauge theory. By exploiting a generalized duality technique, we show that the 3D toric code exhibits optimal thresholds of $p^{X,M}_{th} \approx 11\%$ and $p^{Z,M}_{th} \approx 2\%$ against bit-flip and phase-flip errors, respectively. These threshold values show modest reductions compared to the case of perfect measurements, establishing the robustness of the 3D toric code against measurement errors. Our results constitute a substantial advance towards assessing the practical performance of 3D topological codes. This contribution is timely and in high demand, as rapid hardware advancements are bringing complex codes into experimental reach. Moreover, our work highlights the interdisciplinary nature of fault-tolerant quantum computation and holds significant interest for quantum information science, high-energy physics, and condensed matter physics.