Optimal Threshold for Fracton Codes and Nearly Saturated Code Capacity in Three Dimensions

TL;DR

The paper addresses the problem of determining fault-tolerance thresholds for 3D fracton codes, with a focus on self-dual codes like the checkerboard code. It adopts a statistical-mechanical mapping to a disordered spin model along the Nishimori line and uses large-scale parallel-tempering simulations, complemented by duality arguments, to extract the optimal code-capacity threshold. The checkerboard code is found to have a threshold $p_{th} \simeq 0.108(2)$, near the theoretical limit of about $0.11$, and the duality relation $H(p_{th}) + H(\tilde{p}_{th}) \approx 1$ is validated for fracton codes; Haah's code is conjectured to share the near-limit threshold as well. The work demonstrates fracton codes as highly resilient quantum memories, provides a practical benchmark for decoders, and shows that generalized duality can substantially reduce computational effort in threshold studies.

Abstract

Fracton codes have been intensively studied as novel topological states of matter, yet their fault-tolerant properties remain largely unexplored. Here, we investigate the optimal thresholds of self-dual fracton codes, in particular the checkerboard code, against stochastic Pauli noise. By utilizing a statistical-mechanical mapping combined with large-scale parallel tempering Monte Carlo simulations, we calculate the optimal code capacity of the checkerboard code to be $p_{th} \simeq 0.108(2)$. This value is the highest among known three-dimensional codes and nearly saturates the theoretical limit for topological codes. Our results further validate the generalized entropy relation for two mutually dual models, $H(p_{th}) + H(\tilde{p}_{th}) \approx 1$, and extend its applicability beyond standard topological codes. This verification indicates the Haah's code also possesses a code capacity near the theoretical limit $p_{th} \approx 0.11$. These findings highlight fracton codes as highly resilient quantum memory and demonstrate the utility of duality techniques in analyzing intricate quantum error-correcting codes.

Figures (5)

• Figure 1: Illustration of the checkerboard code and its excitations. (a) Physical qubits (black dots) reside on the vertices of the lattice. Stabilizers $\hat{A}_c$ and $\hat{B}_c$ are defined on each colored cube $c$ as products of $\hat{\sigma}^x$ and $\hat{\sigma}^z$ operators, respectively, acting on the vertex set $\partial c$. (b) Fracton excitations are generated in quartets by a rigid one-dimensional string operator. (c, d) A dipolar pair of fractons with even separation moves freely in the plane perpendicular to its dipole moment.
• Figure 2: Illustration of the statistical mapping from the checkerboard code (left) to the tetrahedral Ising model on a face-centered cubic lattice (right) derived from Canossa24. Blue and cyan circles, located at the vertices and face centers of the FCC lattice, represent the Ising spins associated with each stabilizer, constructing the dual lattice of the checkerboard code. The action of a single qubit error is mapped into a four-body interaction with strength $\eta_{\mathbf{v}}^{\pm}$; these are defined for each lattice site $\mathbf{v}$ in the FCC lattice and are represented by a red (green) tetrahedron placed in each shaded cube (blank cube) of the lattice on the right.
• Figure 3: Energy histograms at the transition temperature for a representative disorder value below threshold ($p = 0.090$). While a sharpening of the double-peak structure with increasing system size is consistent with a first-order transition, it can also occur if the two peaks move closer together as $L$ grows, reflecting strong finite-size effects. The decisive signature of a first-order transition is the suppression of energetic configurations between the peaks, which is expected to become more pronounced with increasing system size (as shown in the inset). This provides a reference for the expected behavior below the error threshold.
• Figure 4: Disorder-averaged energy Histograms (top row) and correlation lengths (bottom row) at different disorder values and for various system sizes. While the energy histograms present a typical first-order double-peak behavior at all disorder values due to strong finite-size effects, the presence of a first-order transition at $p=0.105$ is made obvious by the sharpening of the two peaks and a deepening of the minimum between them as the system size grows, a feature that can still be observed, although in less evident capacity, at $p=0.107$. This lowering of the minimum corresponds to a growing domain wall free energy cost as the system size increases which, in the thermodynamic limit, determines the presence of a first-order transition. On the other hand, at $p=0.110$ the minimum goes up with system size, indicating that the first-order features of the system are slowly disappearing. This is further highlighted by the sharp absence of any crossing in the correlation lengths regardless of system size, another indicator of a crossover regime. The sharp contrast between these behaviours allows us to identify the ordered and disordered phases of the model, and estimate the optimal error threshold to sit in between $0.107$ and $0.110$.
• Figure 5: (a) Illustration of the stabilizer structure of the Haah's code. Two sets qubits are situated at the vertices of a cubic lattice. The $X-$ and $Z-$ type stabilizers $A_C,B_C$ are defined for each unit cell $C$ as a product of Pauli operators acting on a subset of the 16 qubits residing at the vertices of the cube. (b) A single-qubit $X$ (or $Z$) error excites 4 cubic stabilizer operators: since the structures are equivalent for $X$ and $Z$ stabilizers up to a global rotation, both independent noise models can be represented by random the Fractal Ising Model Eq. \ref{['eq:FIM_model']}.