Self-Dual Fracton Codes

• Self-dual fracton codes are three-dimensional stabilizer models characterized by an exact symmetry between X- and Z-type errors, yielding optimal error thresholds.
• They map quantum error configurations to classical Ising models with tetrahedral and fractal interactions, where sub-dimensional symmetries dictate phase behavior.
• Key findings include extensive ground-state degeneracy and an error threshold near 0.108, making these codes highly fault-tolerant for quantum memory applications.

Self-dual fracton codes are a class of three-dimensional stabilizer codes characterized by immobile point excitations ("fractons") and an exact symmetry between $X$-type and $Z$-type errors, manifested as self-duality. These codes, including the checkerboard code (type-I) and Haah's code (type-II), excel in quantum memory applications due to their high ground-state degeneracy (GSD), sharply defined error thresholds, and fault-tolerance properties. The mathematical structure underlying these codes is closely related to exotic Ising spin models—the tetrahedral Ising model (TIM) for the checkerboard code and the fractal Ising model (FIM) for Haah’s code—where sub-dimensional or fractal symmetries dictate the nature of ordering and phase transitions. Recent research elucidates their optimal thresholds against stochastic Pauli noise and reveals anomalous finite-size effects connected to their underlying symmetry breaking (Canossa et al., 28 Dec 2025, Canossa et al., 2023).

1. Theoretical Foundations and Stabilizer Structures

Fracton codes are defined on three-dimensional lattices where each qubit is situated at a vertex. For self-dual codes, the stabilizer generators partition into two sublattices associated with Pauli $X$ and $Z$ operations. In the checkerboard code, cubes are organized in a checkerboard pattern: "A" cubes host $X$-type stabilizers ($A_c = \prod_{i\in\partial c} \sigma_i^x$) and alternating "B" cubes host $Z$-type stabilizers ($B_c = \prod_{i\in\partial c} \sigma_i^z$), with $\partial c$ denoting the vertices of cube $c$. Self-duality implies that exchanging $X \leftrightarrow Z$ maps the code onto itself, rendering the $X$ and $Z$ error problems isomorphic (Canossa et al., 28 Dec 2025).

Haah’s code generalizes this structure by assigning two qubits per vertex and placing two 8-body stabilizers per cube. Any single-qubit error triggers a fractal cluster of four stabilizer violations, highlighting the code’s fractal architecture (Canossa et al., 28 Dec 2025).

2. Ising Model Mapping and Duality

Error configurations in these codes correspond to classical spin models. Specifically, the decoding problem maps Pauli-based quantum errors to disorder in a classical Ising Hamiltonian on the face-centered cubic (FCC) lattice for the checkerboard code:

$$H_{\rm Tetra} = -\sum_{i} \eta_i\,\sigma_{c(i,1)} \sigma_{c(i,2)} \sigma_{c(i,3)} \sigma_{c(i,4)}$$

where $\eta_i$ encodes the error (with $\eta_i = \pm 1$) and each qubit touches four stabilizer centers $c$.

For Haah’s code, the mapping leads to a random fractal Ising Hamiltonian with two interlaced 4-body interactions per cube for each qubit flavor:

$$H_{\rm FIM} = -\sum_{\mathbf v}\left[ \eta^+_{\mathbf v}S^+_{\mathbf v}+\eta^-_{\mathbf v}S^-_{\mathbf v} \right]$$

where $S^\pm_{\mathbf v}$ generates fractal spin interactions. In both cases, decoding is equivalent to detecting an ordered phase in the corresponding spin model (Canossa et al., 28 Dec 2025, Canossa et al., 2023).

Kramers–Wannier self-duality is exact: dual spins are placed at the centers of tetrahedral interactions, and under duality, the Hamiltonian is invariant up to a relabeling of interactions. The self-dual point is found at $\beta_c = \frac{1}{2} \ln(\sqrt{2}+1)$ (Canossa et al., 2023).

3. Symmetry Generators and Ground-State Degeneracy

Self-dual fracton codes exhibit sub-dimensional or fractal symmetry generators. In the TIM (checkerboard code), planar-flip symmetries enable simultaneous flipping of all spins in a given plane (e.g., all spins in the $xy$-plane at $x=i$):

• There are $3L$ independent plane-flip generators minus three global constraints, so $\log_2$ GSD $= 3L-3$.

In the FIM (Haah's code), symmetries are encoded using elements of the polynomial ring $R=\mathbb F_2[x,y,z]/(x^L-1, y^L-1, z^L-1)$. The kernel of a linear map induced by the interactions yields $2^{2L-1}$ independent fractal symmetries for system sizes $L=2^n$, and thus GSD$(H_{FIM}) = 2^{2L-1}$ (Canossa et al., 2023).

These unconventional symmetry generators underlie the codes’ extensive ground-state degeneracy and exotic order parameters.

4. Non-local and Fractal Order Parameters

The conventional local magnetization does not survive in the presence of sub-dimensional or fractal symmetries. Non-local order parameters are thus constructed. In the TIM, extended semi-local order is characterized by correlators spanning multiple planes:

$$G^z_{TIM}(r) = (8/L^3) \sum_v \left[ s_v s_{v+\hat{z}+\hat{z}} s_{v+\hat{y}+r\hat{z}} s_{v+\hat{z}+r\hat{z}} \right]$$

and the sub-dimensional order parameter,

$$Q^z_{TIM} = (4/L^3) \sum_{x,y} \left| \sum_z [ s_v s_{v+\hat{x}+\hat{y}} + s_{v+\hat{y}+\hat{z}} s_{v+\hat{x}+\hat{z}} ] \right|$$

encapsulates line-wise correlations.

In the FIM, fractal order correlators are constructed: $$G_{FIM}(r) = (1/L^3)\sum_v s_v\,s_{v+r\hat{x}}\,s_{v+r\hat{y}}\,s_{v+r\hat{z}}$$ Here $r=2^n$ ensures the support spans a fractal cluster; ordered phases exhibit $G_{FIM}(r)\to 1$ as $r\to\infty$. These order parameters diagnose spontaneous breaking of sub-dimensional or fractal symmetries (Canossa et al., 2023).

5. Phase Transitions and Finite-Size Effects

Both the tetrahedral and fractal Ising models (TIM, FIM) undergo strong first-order phase transitions at $\beta_c = \frac{1}{2}\ln(\sqrt{2}+1)$. Canonical energy histograms manifest double peaks separated by a barrier that increases with system size. Binder cumulant analyses reveal negative dips, and specific-heat and susceptibility diverge at the transition.

The models display anomalous finite-size scaling: The shift of the critical point with system size $L$ is $\sim 1/L^2$, in contrast to the $1/L^3$ shift typical for non-degenerate three-dimensional systems. This scaling results from sub-extensive degeneracy $GSD \sim 2^{\alpha L^d}$ with $d=1$ for both TIM (plane-flip) and FIM (fractal symmetry), as confirmed by Monte Carlo simulations for $L$ up to 28 (TIM) and $L=\{8,16\}$ (FIM) (Canossa et al., 2023).

6. Code Capacity, Thresholds, and Fault-Tolerance

The error-correction threshold for the checkerboard code is computed to be $p_{th} \simeq 0.108(2)$, close to the theoretical optimum $p_{th} \approx 0.11$, representing the highest known threshold among three-dimensional topological codes. This threshold nearly saturates the quantum Gilbert–Varshamov bound for topological codes.

A generalized entropy duality holds: for self-dual codes, $2H(p_{th}) \approx 1$ where $H(p)$ is the binary Shannon entropy, numerically $2H(0.108) \approx 0.987(8) \approx 1$. This duality extends to other fracton codes, including Haah’s code, implying similar near-optimality. The first-order nature of the transition guarantees a sharply defined threshold: logical error rates jump rapidly for $p>p_{th}$ and vanish for $p<p_{th}$ (Canossa et al., 28 Dec 2025).

7. Implications and Perspectives

Self-dual fracton codes such as the checkerboard and Haah’s codes are distinguished by their resilience as quantum memories, possessing thresholds comparable to leading 2D codes but realized in three dimensions. The utility of duality techniques allows for threshold estimates for complex codes with minimal computational overhead. A plausible implication is the applicability of these methods to subsystem-symmetric and fractal codes beyond the cases studied, including bicycle codes.

The identification of sub-dimensional and fractal order parameters, together with the understanding of their finite-size scaling and ground-state degeneracy, provides a robust framework for the analysis of quantum error-correction in intricate stabilizer codes. Continued study of the disorder–temperature phase diagram and scaling properties is likely to further clarify the ultimate fault-tolerance capabilities of these codes (Canossa et al., 28 Dec 2025, Canossa et al., 2023).