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arXiv:2602.23751 · Physical Review A

Spin stiffness and resilience phase transition in a noisy toric-rotor code

Morteza Zarei, Mohammad Hossein Zarei

quant-phcond-mat.stat-mechSubmitted Feb 27, 2026• Announced Mar 2, 2026• 0 citations• Revised Aug 6, 2026 (v2)
v1 · Feb 27, 2026v2 · Aug 6, 2026
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Abstract

We use a quantum formalism for the partition function of the classical $XY$ model to identify a resilience phase transition in the zero-syndrome postselected sector of a noisy toric-rotor code. To this end, we consider a logical state of toric-rotor code under phase-shift noise described by a von Mises probability distribution. We then show that the fidelity of the noisy state with respect to the initial logical state is proportional to the partition function of the $XY$ model, such that a Kosterlitz-Thouless phase transition at a critical temperature $T_c$ corresponds to a resilience phase transition at a critical width $σ_c$. To characterize this transition, we map the spin stiffness of the $XY$ model to a topological order parameter $0\leq λ\leq 1$, which quantifies the intrinsic resilience of the code to decoherence within the zero-syndrome subspace. We show that the initial logical state exhibits partial resilience to noise for widths less than $σ_c \approx 0.89$, where $λ$ satisfies $0< λ<1$ and drops discontinuously to zero at $σ_c$. We further discuss the implications of our results for postselected quantum error correction in the toric-rotor code in higher dimensions. Our work shows that the quantum formalism for partition functions provides a mathematically rigorous framework for studying noisy continuous-variable quantum codes.

Journal reference
Phys. Rev. A 114, 022440 (2026)
DOI
10.1103/q4zh-9wr4
Comments
11 pages, 7 figures, Accepted for publication in Physical Review A

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We use a quantum formalism for the partition function of the classical XYXYXY model to identify a resilience phase transition in the zero-syndrome postselected sector of a noisy toric-rotor code. To this end, we consider a logical state of toric-rotor code under phase-shift noise described by a von Mises probability distribution. We then show that the fidelity of the noisy state with respect to the initial logical state is proportional to the partition function of the XYXYXY model, such that a Kosterlitz-Thouless phase transition at a critical temperature TcT_cTc​ corresponds to a resilience phase transition at a critical width σcσ_cσc​. To characterize this transition, we map the spin stiffness of the XYXYXY model to a topological order parameter 0≤λ≤10\leq λ\leq 10≤λ≤1, which quantifies the intrinsic resilience of the code to decoherence within the zero-syndrome subspace. We show that the initial logical state exhibits partial resilience to noise for widths less than σc≈0.89σ_c \approx 0.89σc​≈0.89, where λλλ satisfies 0<λ<10< λ<10<λ<1 and drops discontinuously to zero at σcσ_cσc​. We further discuss the implications of our results for postselected quantum error correction in the toric-rotor code in higher dimensions. Our work shows that the quantum formalism for partition functions provides a mathematically rigorous framework for studying noisy continuous-variable quantum codes.
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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.
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Quantum error correction protects quantum information against decoherence, provided the noise strength remains below a critical threshold. This threshold marks the critical point for the decoding phase transition. Here, we connect this transition in the toric code to a topological phase transition in disordered Majorana fermions at high temperatures. A quantum memory in the error-correctable phase is captured by the presence of a Majorana zero mode, trapped in vortex defects associated with twisted boundary conditions. These results are established by expressing the coherent information, which measures the amount of recoverable quantum information in a given noisy code, in terms of a mixed-state topological order parameter of fermions. Our work hints at a broader connection of the robustness of quantum information in stabilizer codes and mixed-state topological phase transitions in symmetry-protected fermion matter.
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