Subsystem Symmetries and Fracton Models in Quantum Error Correction

Abstract: Constructing new quantum codes and understanding their error resilience are central challenges in the development of robust quantum memories. Topological codes are particularly promising due to their favorable error-correcting properties and their connections to phases of matter in many-body physics. In this thesis, we explore the interplay between classical Ising models and quantum error correction through subsystem symmetries, fracton topological order, and Kramers-Wannier-type duality. We study two three-dimensional classical self-dual Ising models with subsystem symmetries, the Tetrahedral Ising model and the Fractal Ising model, investigating their thermal behavior, their relation to fracton phases through subsystem-symmetry gauging, and the properties of the resulting fracton codes. Using a statistical-mechanical mapping, we determine the optimal code-capacity threshold of the Checkerboard code to be $0.107(3)$, which saturates the theoretical limit for CSS codes and represents the highest optimal error threshold among known three-dimensional codes. We relate this saturation to a generalized entropy relation for classical spin models satisfying a Kramers-Wannier-type duality, and argue how this prediction extends to CSS codes with zero encoding rate whose $X$- and $Z$-noise models map to classically dual spin models. These findings establish fracton codes as highly resilient candidates for quantum memories and demonstrate the power of the statistical-mechanical framework, together with its duality predictions, in analyzing and constructing robust quantum error-correcting codes.

• The paper identifies a code-capacity threshold of 0.107(3) for the Checkerboard fracton code, nearly saturating the Hamming bound and highest among known three-dimensional CSS codes.
• The dissertation reveals that subsystem symmetry breaking in 3D Ising models leads to stranger-than-standard scaling behavior.
• An essential result relates rigorously the error-correction threshold and thermodynamic transitions, highlighting the statistical-mechanical nature of code thresholds and performance.

Overview

This dissertation establishes a unified connection between classical Ising models with subsystem symmetries, fracton topological order obtained by gauging those symmetries, and the performance of the resulting quantum codes as error-correcting memories. The central quantitative result is an optimal code-capacity threshold of $p^{\text{th}}=0.107(3)$ for the Checkerboard fracton code under independent Pauli noise — a value that nearly saturates the theoretical bound $H(p_X)+H(p_Z)\leq 1$ and constitutes the highest optimal threshold among known three-dimensional CSS codes. The work proceeds in three stages: a numerical characterization of two classical subsystem-symmetric self-dual Ising models (the Tetrahedral and Fractal Ising models), the derivation of their gauged fracton counterparts (the Checkerboard model and Haah's cubic code), and a statistical-mechanical analysis of the decoding problem that ties code thresholds to duality properties of disordered Ising models.

Classical subsystem-symmetric Ising models

The first part studies two three-dimensional classical spin models whose Hamiltonians possess subextensive sets of symmetries acting on lower-dimensional supports. The Tetrahedral Ising model (TIM) is defined on a face-centered cubic lattice with four-spin tetrahedral interactions and exhibits $3L-3$ independent planar $\mathbb{Z}_2$ subsystem symmetries generated by plane flips. The Fractal Ising model (FIM) lives on a simple cubic lattice with two distinct tetrahedral couplings per unit cell and possesses fractal subsystem symmetries, constructed explicitly via the polynomial ring formalism over $\mathbb{Z}_2[x,y,z]$. A key structural result is that the FIM's symmetry group — and hence its ground-state degeneracy — depends discontinuously on system size: for $L=2^n$ the degeneracy is $2^{2L-1}$, while other families such as $L=4^n-1$ yield $2^{2L-5}$ and certain classes have only constant degeneracy.

Both models are shown to be self-dual under a generalized Kramers–Wannier (Wegner) duality, which fixes their thermodynamic transition point at the self-dual value $\beta_c = \frac{1}{2}\ln(\sqrt{2}+1)$. Multicanonical Monte Carlo simulations demonstrate that both models exhibit a single strongly first-order phase transition separating a disordered phase from a phase with planar or fractal long-range order. The finite-size shift of the pseudocritical temperature follows the anomalous scaling $H(p_X)+H(p_Z)\leq 1$0 with $H(p_X)+H(p_Z)\leq 1$1, consistent with simultaneous breaking of all subsystem symmetries and the resulting subextensive ground-state degeneracy $H(p_X)+H(p_Z)\leq 1$2. This non-standard scaling — verified numerically against competing $H(p_X)+H(p_Z)\leq 1$3 and $H(p_X)+H(p_Z)\leq 1$4 fits for the FIM — is identified as a hallmark of subsystem symmetry breaking.

The dissertation is careful to state what these results do not establish: all known 3D subsystem-symmetric Ising and compass models exhibit first-order transitions, but no theorem enforces this, dimensional-reduction bounds do not constrain transition order, and macroscopic degeneracy alone does not forbid continuous transitions (as anisotropic layer limits show conventional critical behavior). Whether an isotropic short-range 3D subsystem-symmetric model can undergo a continuous transition into subsystem order remains open.

Gauging and fracton topological order

The second part develops the generalized gauging procedure for subsystem symmetries. Starting from a transverse-field quantum version of a subsystem-symmetric Ising model, introducing nexus fields at each multispin coupling, and imposing local Gauss law constraints yields $H(p_X)+H(p_Z)\leq 1$5 lattice gauge theories whose excitation structure is dictated by the geometry of the ungauged symmetries. Three algebraic conditions govern whether the gauged theory is topologically ordered with immobile charges: a codimension condition (via the Buchsbaum–Eisenbud criterion) ensuring local indistinguishability, a subsymmetry-dimensionality condition excluding symmetries on manifolds of dimension below $H(p_X)+H(p_Z)\leq 1$6, and a fracton condition forbidding binomial interaction terms.

Gauging the planar symmetries of the TIM produces the Checkerboard model, a CSS stabilizer code with eight-body cube stabilizers arranged in a checkerboard pattern, encoding $H(p_X)+H(p_Z)\leq 1$7 logical qubits with distance $H(p_X)+H(p_Z)\leq 1$8. Its excitations are strictly immobile individual fractons constrained by one charge conservation law per coordinate plane, while fracton dipoles exhibit constrained motion via string and membrane operators; distance-2 staggered membranes reproduce the braiding structure of the 2D toric code, reflecting a "topological dimensional reduction." Because rigid string-like logical operators exist, the energy barrier against logical failure is system-size independent, so Checkerboard topological order cannot persist at any nonzero temperature — a limitation shared by the toric code and all type-I fracton models.

Gauging the fractal symmetries of the FIM yields Haah's cubic code, encoding $H(p_X)+H(p_Z)\leq 1$9 qubits (for $3L-3$0). Its logical operators have fractal support determined by the Sierpiński rule, with weight scaling superlinearly, $3L-3$1, $3L-3$2. The energy barrier grows only logarithmically, $3L-3$3, which yields a memory lifetime growing polynomially as $3L-3$4 — but only below a temperature-dependent cutoff size $3L-3$5, beyond which entropic proliferation dominates and lifetime saturates. Haah's code therefore does not sustain topological order at finite temperature in the thermodynamic limit, though it remains qualitatively more robust than any string-operator code.

Statistical-mechanical mapping and the duality-based threshold relation

The final part formulates maximum-likelihood decoding of CSS codes under i.i.d. Pauli noise as the partition function of a random-bond Ising (RBI) model, with stabilizers mapped to spins, qubits to couplings, and decodability to the existence of an ordered phase containing the Nishimori line. For homological CSS codes, the boundary maps of the underlying chain complex coincide with the incidence matrices of mutually dual Ising models satisfying Wegner's closure and completeness conditions.

Using the replica trick along the Nishimori line, the dissertation derives a principal-factor fixed-point condition for mutually dual disordered Ising models, yielding the prediction

$3L-3$6

which coincides with the upper limit of the quantum Hamming bound $3L-3$7. This explains why known topological CSS codes systematically nearly saturate the bound: their $3L-3$8- and $3L-3$9-noise models map to dual Ising systems. The approximation relies on dominance of the $\mathbb{Z}_2$0 replica sector and implicitly assumes no spin-glass phase or replica symmetry breaking near the multicritical point; the author acknowledges this as an assumption rather than a proven property.

Large-scale parallel-tempering simulations of the random-bond TIM — consuming over $\mathbb{Z}_2$1 CPU hours across disorder realizations at sizes $\mathbb{Z}_2$2 — locate the tricritical point via three diagnostics: sharpening versus softening of double-peaked energy histograms, scaling of the domain-wall free-energy cost ($\mathbb{Z}_2$3 scaling persisting below threshold), and presence versus absence of crossings in the second-moment correlation length. At $\mathbb{Z}_2$4 first-order signatures persist; at $\mathbb{Z}_2$5 the absence of correlation-length crossings rules out a continuous transition, indicating crossover; at $\mathbb{Z}_2$6 softened but persistent ordered-phase signatures remain. Combining these regimes gives $\mathbb{Z}_2$7, corresponding to $\mathbb{Z}_2$8 — near saturation of the bound, as predicted.

For Haah's code, a direct finite-size scaling analysis is precluded because fractal symmetries exist only for $\mathbb{Z}_2$9, strong finite-size effects exclude $\mathbb{Z}_2[x,y,z]$0, and computational cost excludes $\mathbb{Z}_2[x,y,z]$1, leaving effectively one usable size. Instead, the dissertation argues heuristically that since the FIM shares the TIM's strongly first-order clean-limit transition, comparable or larger domain-wall free-energy costs, and no evidence of glassy phases, it should likewise satisfy the principal-factor condition — leading to the conjecture that Haah's code also has a code-capacity threshold near 0.11. This remains unverified numerically.

Limitations and open questions

Several caveats bear directly on the results. First, the duality-based fixed-point condition is derived under a principal-factor approximation whose validity is empirical, supported by consistency across known codes but not established as a theorem; its accuracy is plausibly tied to the absence of replica symmetry breaking along the Nishimori line, which itself is assumed rather than demonstrated for the FIM. Second, the Checkerboard threshold estimate rests on a minimal dataset of three disorder values and small system sizes ($\mathbb{Z}_2[x,y,z]$2); the uncertainty assignment assumes a symmetric plausible interval, and distinguishing a genuinely softened first-order transition from a weakly continuous one at $\mathbb{Z}_2[x,y,z]$3 is not fully resolved. Third, all thresholds are computed under idealized static i.i.d. noise; whether fractonic mobility constraints confer advantage under circuit-level noise — where decoding maps to an effective 4D disordered Ising model — remains computationally out of reach and open. Fourth, although the Checkerboard model admits transversal CNOT, Hadamard, and phase gates, the Eastin–Knill theorem forbids a universal transversal gate set, and locality-preserving operations act collectively on the full encoded subspace, preventing independent addressing of individual logical qubits; overcoming this would require modified boundaries whose effect on threshold and rate is unknown.

Conclusion

This work demonstrates that subsystem-symmetric Ising models, fracton gauge theories, and their associated CSS codes share a common statistical-mechanical backbone: gauging maps classical symmetries to quantum Gauss laws, and the SM mapping inverts this construction up to quenched disorder, placing code thresholds within the framework of duality in disordered spin systems. The measured Checkerboard threshold of 0.107(3), combined with an encoding rate six times that of the 2D toric code, identifies fracton codes as competitive candidates for fault-tolerant quantum memory under idealized noise. The broader significance lies in the duality-based saturation mechanism, which converts an otherwise prohibitive threshold computation into a tractable fixed-point condition and predicts near-optimal performance for Haah's code — a prediction awaiting direct numerical confirmation.