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

TL;DR

The paper analyzes the 3D toric code under Pauli and measurement noise by mapping error correction to higher-dimensional, disordered lattice gauge theories. It derives two coupled statistical-mechanical models (a 4D random-plaquette Z2 gauge model and a 4D random-cube Z2 gauge model) and applies generalized duality along the Nishimori line to calculate optimal fault-tolerance thresholds. The results show phenomenological thresholds of $p^{X,M}_{th} \approx 11\%$ and $p^{Z,M}_{th} \approx 2\%$, with the overall threshold set by the smaller; results indicate robustness against measurement errors and provide analytic benchmarks for 3D topological codes. The work connects quantum error correction with higher-form gauge theories and duality methods, offering insights for decoding performance and guiding future explorations of 3D codes and related gauge-theoretic frameworks.

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.

Figures (8)

• Figure 1: Stabilizers of the 3D toric code. (a) Each link of the lattice hosts one physical qubit. The vertex operator $A_s$ (red) acts on the six qubits located on the links incident to vertex $s$. The plaquette operator $B_p$ (blue) acts on the four qubits surrounding plaquette $p$. (b) The same operators viewed on the dual lattice.
• Figure 2: Logical operators of the 3D toric code. The logical operator $\bar{Z}_i$ (blue) is a product of Pauli-$Z$ operators along a noncontractible string that winds around the lattice. The logical operator $\bar{X}_i$ (red) is a product of Pauli-$X$ operators acting on the links intersected by a noncontractible membrane on the dual lattice. These two logical operators anticommute because the string and membrane intersect at a single qubit.
• Figure 3: Geometric picture of errors and syndromes. (a) A spatial lattice showing a syndrome configuration (black dots) and two possible $Z$-error chains, $\mathcal{E}$ (red links) and $\mathcal{E}'$ (blue links), that both lead to this syndrome. (b) A spacetime lattice showing a syndrome configuration $\mathcal{S}$ (black temporal links) obtained from faulty measurements of the $A$-type stabilizers, together with a compatible error chain $\mathcal{E}$ that includes both $Z$ errors (red spatial links) and measurement faults (red temporal links).
• Figure 4: Illustration of 4D $\mathbb{Z}_2$ lattice gauge models on a tesseract. (a) Random-Plaquette Gauge Model (RPGM): This model is defined by $\mathbb{Z}_2$ spin variables $\sigma_\ell =\pm 1$ residing on the links $\ell$ of the hypercubic lattice. The Hamiltonian involves a four-body interaction on plaquettes $p$, with a random coupling sign $\eta_p =\pm 1$. (b) Random-Cube Gauge Model (RCGM): This model is defined on the dual lattice. The spin variables $\sigma_{p^*}=\pm 1$ reside on each plaquette $p^*$, and a six-body interaction with random coupling sign $\eta_{c^*}=\pm 1$ is located on each 3D cubes $c^*$.
• Figure 5: Connections between topological codes, statistical mechanics, and thresholds. This schematic outlines the statistical-mechanical mapping of a topological CSS code to its effective and dual spin models, linked by generalized and Kramers-Wannier dualities, and ultimately to optimal thresholds through an entropy relation.