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arXiv:2610.37791

Topological Quantum Error-Correcting Thresholds of a Symmetry-Enriched Toric Code

Jiasheng He, Ryan Thorngren

quant-phcond-mat.dis-nncond-mat.stat-mechSubmitted Oct 27, 2026• Announced Oct 29, 2026• 0 citations
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Abstract

We study the error thresholds of a new topological quantum error-correcting code under the independent bit-flip and phase-flip noisy channel. By mapping to classical spin-glass models, we find that if Pauli X and Z errors are equally probable, the threshold of vertex syndromes is improved by 30% compared to that of the toric code, while the threshold of plaquette syndromes is lowered. Both thresholds increase when the Z component is less likely than the X component, reaching 0.5 if the Z component is absent. An analytical proof justifies this special case of perfect correctability, and a disordered version of Peierls's argument establishes a lower bound for the vertex-syndrome threshold.

AI Summary

This paper studies ideal-syndrome error thresholds of a crystalline symmetry-enriched toric code with qubits on links and vertices, under independent $X$- and $Z$-error components of probabilities $p$ and $q$. The code is finite-depth Clifford-equivalent to the toric code with ancillary qubits, but its vertex syndromes reveal corners as well as endpoints of error strings. The two syndrome-decoding problems map to different disordered classical models: an Ising model with random nearest-neighbor and four-spin plaquette interactions, and two perpendicular layers of Ising chains with interlayer coupling. Thresholds are identified with ordering transitions on a Nishimori surface using parallel-tempering Monte Carlo and transfer-matrix domain-wall free energies. At $p=q$, the reported vertex threshold is $p_N\approx0.14211448$, with a statistical 95% interval $(0.14210878,0.14211952)$, versus approximately $0.1094$ for the toric code. The plaquette threshold instead lies between $0.083$ and $0.09$, limiting the full code under symmetric noise. At fixed $q=0.04$, the corresponding thresholds are approximately $0.2159$ and $0.26$–$0.30$. A disordered Peierls argument gives the vertex-sector lower bound $p_N>0.0654$ at $p=q$. For $q=0$, both thresholds are reported to reach $0.5$; the vertex-sector argument uses an effective one-dimensional model, while the plaquette-sector argument uses mean field. Syndrome measurements are assumed perfect. Plaquette thresholds remain coarsely resolved, and transfer-matrix threshold estimates use widths only $6,8,10$. Measurement noise, depolarizing noise, possible spin-glass behavior, and reentrance are left for further study.

Little sign of AI-generated textOverclaimsStrong assumptionsMissing comparison

Central claim may not hold: The Peierls proof incorrectly treats contour-corner energies as additive when contours share a plaquette, invalidating the supplied derivation of the claimed 0.0654 vertex-threshold lower bound. The q=0 vertex proof incorrectly discards one transfer-matrix contribution: its two partition-function weights approach 1/2 each, rather than 1 and 0. The claimed analytical proof of q=0 perfect correctability is incomplete for plaquette syndromes, where the paper supplies only a mean-field argument.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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We study the error thresholds of a new topological quantum error-correcting code under the independent bit-flip and phase-flip noisy channel. By mapping to classical spin-glass models, we find that if Pauli X and Z errors are equally probable, the threshold of vertex syndromes is improved by 30% compared to that of the toric code, while the threshold of plaquette syndromes is lowered. Both thresholds increase when the Z component is less likely than the X component, reaching 0.5 if the Z component is absent. An analytical proof justifies this special case of perfect correctability, and a disordered version of Peierls's argument establishes a lower bound for the vertex-syndrome threshold.
This paper studies ideal-syndrome error thresholds of a crystalline symmetry-enriched toric code with qubits on links and vertices, under independent XXX- and ZZZ-error components of probabilities ppp and qqq. The code is finite-depth Clifford-equivalent to the toric code with ancillary qubits, but its vertex syndromes reveal corners as well as endpoints of error strings.
The two syndrome-decoding problems map to different disordered classical models: an Ising model with random nearest-neighbor and four-spin plaquette interactions, and two perpendicular layers of Ising chains with interlayer coupling. Thresholds are identified with ordering transitions on a Nishimori surface using parallel-tempering Monte Carlo and transfer-matrix domain-wall free energies. At p=qp=qp=q, the reported vertex threshold is pN≈0.14211448p_N\approx0.14211448pN​≈0.14211448, with a statistical 95% interval (0.14210878,0.14211952)(0.14210878,0.14211952)(0.14210878,0.14211952), versus approximately 0.10940.10940.1094 for the toric code. The plaquette threshold instead lies between 0.0830.0830.083 and 0.090.090.09, limiting the full code under symmetric noise. At fixed q=0.04q=0.04q=0.04, the corresponding thresholds are approximately 0.21590.21590.2159 and 0.260.260.26–0.300.300.30. A disordered Peierls argument gives the vertex-sector lower bound pN>0.0654p_N>0.0654pN​>0.0654 at p=qp=qp=q.
For q=0q=0q=0, both thresholds are reported to reach 0.50.50.5; the vertex-sector argument uses an effective one-dimensional model, while the plaquette-sector argument uses mean field. Syndrome measurements are assumed perfect. Plaquette thresholds remain coarsely resolved, and transfer-matrix threshold estimates use widths only 6,8,106,8,106,8,10. Measurement noise, depolarizing noise, possible spin-glass behavior, and reentrance are left for further study.
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 Z 2 gauge theory. By exploiting a generalized duality technique, we show that the 3D toric code exhibits optimal thresholds of p th X , M ≈ 11 % and p th Z , M ≈ 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.
Topological quantum error correction codes are currently among the most promising candidates for efficiently dealing with the decoherence effects inherently present in quantum devices. Numerically, their theoretical error threshold can be calculated by mapping the underlying quantum problem to a related classical statistical-mechanical spin system with quenched disorder. Here, we present results for the general fault-tolerant regime, where we consider both qubit and measurement errors. However, unlike in previous studies, here we vary the strength of the different error sources independently. Our results highlight peculiar differences between toric and color codes. This study complements previous results published in New J. Phys. 13, 083006 (2011).
Fracton models provide examples of novel gapped quantum phases of matter that host intrinsically immobile excitations and therefore lie beyond the conventional notion of topological order. Here, we calculate optimal error thresholds for quantum error correcting codes based on fracton models. By mapping the error-correction process for bit-flip and phase-flip noises into novel statistical models with Ising variables and random multibody couplings, we obtain models that exhibit an unconventional subsystem symmetry instead of a more usual global symmetry. We perform large-scale parallel tempering Monte Carlo simulations to obtain disorder-temperature phase diagrams, which are then used to predict optimal error thresholds for the corresponding fracton code. Remarkably, we found that the X-cube fracton code displays a minimum error threshold (7.5%) that is much higher than 3D topological codes such as the toric code (3.3%), or the color code (1.9%). This result, together with the predicted absence of glass order at the Nishimori line, shows great potential for fracton phases to be used as quantum memory platforms.
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.
The inevitable presence of decoherence effects in systems suitable for quantum computation necessitates effective error-correction schemes to protect information from noise. We compute the stability of the toric code to depolarization by mapping the quantum problem onto a classical disordered eight-vertex Ising model. By studying the stability of the related ferromagnetic phase via both large-scale Monte Carlo simulations and the duality method, we are able to demonstrate an increased error threshold of 18.9(3)% when noise correlations are taken into account. Remarkably, this result agrees within error bars with the result for a different class of codes—topological color codes—where the mapping yields interesting new types of interacting eight-vertex models.
It is known that noisy topological quantum codes are related to random-bond Ising models where the order-disorder phase transition in the classical model is mapped to the error threshold of the corresponding topological code. On the other hand, there is a dual mapping between classical spin models and quantum Calderbank-Shor-Steane (CSS) states where the partition function of a classical model defined on a hypergraph HHH is written as an inner product of a product state and a CSS state on dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$. It is then interesting to see what the interpretation is of the classical phase transition in the random-bond Ising model within the framework of the above duality, and whether such an interpretation has any connection to the error threshold of the corresponding topological CSS code. In this paper, we consider the above duality relation specifically for a two-dimensional random-bond Ising model. We show that the order parameter of this classical system is mapped to a coherence order parameter in a noisy Toric code model. In particular, a quantum phase transition from a coherent phase to a noncoherent phase occurs when the initial coherent state is affected by two sequences of bit-flip quantum channels where a quenched disorder is induced by measurement of the errors after the first channel. On the other hand, the above transition is directly related to the error threshold of the Toric code model. Accordingly, and since the noisy process can be applied to other topological CSS states, we conclude that the dual correspondence can also provide a useful tool for the study of error thresholds in different topological CSS codes.