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arXiv:2607.21160

Duality constrains optimal thresholds in quantum error correction

Lucas H. English, Haoyuan Luo, Yangming Wang, Basudha Srivastava, Stephen D. Bartlett, Dominic J. Williamson

quant-phSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations
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Abstract

Error correction thresholds are often treated as the primary figure of merit for comparing quantum error-correcting code families. We show that the optimal error correction threshold for many commonly considered codes is constrained to a single universal value at leading order in a replica limit. Through a statistical mechanical mapping, we demonstrate that duality constrains all zero-rate em-symmetric CSS codes to have the same optimal code capacity threshold. Here, em symmetry means that the X- and Z-type parity-check matrices are equivalent up to row and column permutations. Under this statistical mechanical mapping, em-symmetric CSS codes are self-dual under a generalized Kramers-Wannier duality up to a mixing of logical sectors. For zero-rate code families, this mixing contributes only subextensive corrections, so the thermodynamic bulk free energy is self-dual in the trivial logical sector. This self-duality fixes the clean critical point and constrains the disordered phase boundary. We also show that self-duality is preserved under code concatenation, and that optimal decoding of concatenated codes can be reformulated as a renormalization group flow on a hierarchical lattice. Our results provide a common framework for analyzing topological, concatenated, and more general quantum low-density parity-check code families, including both their optimal code capacity thresholds and their sub-threshold logical error suppression.

Comments
27 pages, comments welcome

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Error correction thresholds are often treated as the primary figure of merit for comparing quantum error-correcting code families. We show that the optimal error correction threshold for many commonly considered codes is constrained to a single universal value at leading order in a replica limit. Through a statistical mechanical mapping, we demonstrate that duality constrains all zero-rate em-symmetric CSS codes to have the same optimal code capacity threshold. Here, em symmetry means that the X- and Z-type parity-check matrices are equivalent up to row and column permutations. Under this statistical mechanical mapping, em-symmetric CSS codes are self-dual under a generalized Kramers-Wannier duality up to a mixing of logical sectors. For zero-rate code families, this mixing contributes only subextensive corrections, so the thermodynamic bulk free energy is self-dual in the trivial logical sector. This self-duality fixes the clean critical point and constrains the disordered phase boundary. We also show that self-duality is preserved under code concatenation, and that optimal decoding of concatenated codes can be reformulated as a renormalization group flow on a hierarchical lattice. Our results provide a common framework for analyzing topological, concatenated, and more general quantum low-density parity-check code families, including both their optimal code capacity thresholds and their sub-threshold logical error suppression.
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)0.107(3)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 XXX- and ZZZ-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.
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Quantum error correction (QEC) codes protect quantum information from errors due to decoherence. Many of them also serve as prototypical models for exotic topological quantum matters. Investigating the behavior of the QEC codes under decoherence sheds light on not only the codes' robustness against errors but also new out-of-equilibrium quantum phases driven by decoherence. The phase transitions, including the error threshold, of the decohered QEC codes can be probed by the systems' R\'enyi entropies SR{S}_{R}SR​ with different R\'enyi indices RRR. In this paper, we study the general construction of the statistical models that characterize the R\'enyi entropies of QEC codes decohered by Pauli noise. We show that these statistical models can be organized into a ``tapestry'' woven by rich duality relations among them. For Calderbank-Shor-Steane (CSS) codes with bit-flip and phase-flip errors, we show that each R\'enyi entropy is captured by a pair of dual statistical models with randomness. For $R=2,3,\ensuremath{\infty}$, there are additional dualities that map between the two error types, relating the critical bit-flip and phase-flip error rates of the decoherence-induced phase transitions in the CSS codes. For CSS codes with an ``ememem symmetry'' between the XXX-type and the ZZZ-type stabilizers, the dualities with $R=2,3,\ensuremath{\infty}$ become self-dualities with super-universal self-dual error rates. These self-dualities strongly constrain the phase transitions of the code signaled by ${S}_{R=2,3,\ensuremath{\infty}}$. For general stabilizer codes decohered by generic Pauli noise, we also construct the statistical models that characterize the systems' entropies and obtain general duality relations between Pauli noise with different error rates.
We investigate how the lattice geometry influences the error thresholds of two-dimensional toric codes. In the presence of bit- or phase-flip errors, the toric code maps onto the two-dimensional random-bond Ising model (RBIM). We determine the critical behaviour of the RBIM using replica-exchange Monte Carlo simulations. While previous studies have explored thresholds for general lattice geometries under suboptimal decoders such as minimum-weight perfect matching, the rapid development of near-optimal decoders makes resolving the ultimate, maximum-likelihood code capacity relevant. We compute these optimal error thresholds on the Nishimori line for the square, honeycomb, triangular, dice, and kagome lattices. Owing to lattice duality, the phase-flip threshold on a given lattice is equivalent to the bit-flip threshold on its dual. We find that the optimal thresholds of dual-lattice pairs display a characteristic duality-driven splitting around the self-dual square-lattice, mirroring the qualitative behaviour observed for suboptimal decoders. While the average coordination number has the strongest impact on the error threshold, our results demonstrate that the detailed arrangement of vertices and plaquettes also plays a significant role in determining its precise value.
We introduce an adaptive simulated annealing algorithm to search for moderately-sized quantum error-correcting codes with high encoding rates and large distances. Our search targets two classes of stabilizer codes: (1) CSS codes, and (2) a subclass of CSS codes that we call ``self-dual with equivalent logicals'' (SWEL) codes, the latter of which which admit transversal implementations of logical Hadamard and phase gates that can be leveraged to construct fault-tolerant gate sets. The search is guided by an energy function that acts as a surrogate for the logical error rate in a code-capacity noise model, combining code distance with a count of minimum-weight logical operators to resolve the discrete plateaux that impede naïve distance optimization. For block lengths of up to 505050 physical qubits, our search finds state-of-the-art CSS and SWEL codes whose distances frequently meet or exceed the variants of the quantum Gilbert-Varshamov bound. In addition to providing favorable seed codes for fault-tolerant architectures based on code concatenation, the codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.
Topologically ordered phases are stable to local perturbations, and topological quantum error-correcting codes enjoy thresholds to local errors. We connect the two notions of stability by constructing classical statistical mechanics models for decoding general Calderbank-Shor-Steane codes and classical linear codes. Our construction encodes correction success probabilities under uncorrelated bit-flip and phase-flip errors, and simultaneously describes a generalized Z 2 lattice-gauge theory with quenched disorder. We observe that the clean limit of the latter is precisely the imaginary-time path integral of the corresponding quantum code Hamiltonian when the errors are turned into a perturbative X or Z magnetic field. Motivated by error-correction considerations, we define general order parameters for all such generalized Z 2 lattice-gauge theories, and show that they are generally lower bounded by success probabilities of error correction. For CSS codes satisfying the low-density parity-check condition and with a sufficiently large code distance, we prove the existence of a low-temperature ordered phase of the corresponding lattice-gauge theories, particularly for those lacking Euclidean spatial locality and/or when there is a nonzero code rate. We further argue that these results provide evidence for stable phases in the corresponding perturbed quantum Hamiltonians, obtained in the limit of continuous imaginary time. To do so, we distinguish space- and timelike defects in the lattice-gauge theory. A high free-energy cost of spacelike defects corresponds to a successful “memory experiment” and suppresses the energy splitting among the ground states, while a high free-energy cost of timelike defects corresponds to a successful “stability experiment” and points to a nonzero gap to local excitations.