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