自旋刚性与韧性相变在噪声托里-旋转码中的研究
Spin stiffness and resilience phase transition in a noisy toric-rotor code
AI总结：
研究通过量子形式化方法揭示托里-旋转码在噪声下的韧性相变，利用XY模型配分函数与逻辑子空间保真度的关系，提出拓扑序参量表征其抗退相干能力。
摘要：
我们利用经典XY模型配分函数的量子形式，识别出噪声托里-旋转码中的韧性相变。具体来说，我们考虑在相位移噪声下（由von Mises概率分布描述）的托里-旋转码，并展示最终状态与初始状态之间的保真度与XY模型的配分函数成正比。我们将XY模型的温度映射到托里-旋转码中噪声的宽度，使得临界温度T_c的Kosterlitz-Thouless相变对应于临界宽度σ_c的混合态相变。为了表征这一相变，我们开发了XY模型中自旋刚性的量子形式，并证明其映射到托里-旋转码逻辑子空间中的门保真度。特别地，我们引入了一个拓扑序参量，用于表征托里-旋转码在逻辑子空间内对退相干的韧性。我们证明逻辑子空间不完全对噪声具有韧性，这是正确性的必要条件。然而，对于宽度小于σ_c≈0.89的情况，韧性序参量接近1，然后在σ_c处降至零。我们还利用这些结果来阐明更高维d>2中托里-旋转码的正确性。我们的工作表明，配分函数的量子形式为研究连续变量量子码的正确性提供了数学上严谨的框架。
英文摘要：
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.
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