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 a noisy toric-rotor code. Specifically, we consider the toric-rotor code under phase-shift noise described by a von Mises probability distribution and show that the fidelity between the final state after noise and the initial state is proportional to the partition function of the XY model. We map the temperature of the XY model to the width of the noise in the toric-rotor code, such that a Kosterlitz--Thouless phase transition at a critical temperature T_c corresponds to a mixed-state phase transition at a critical width σ_c. To characterize this phase transition, we develop a quantum formalism for the spin stiffness in the XY model and show that it is mapped to the gate fidelity in the logical subspace of the toric-rotor code. In particular, we introduce a topological order parameter that characterizes the resilience of the toric-rotor code to decoherence within the logical subspace. We show that the logical subspace does not exhibit complete resilience to noise, which is a necessary condition for correctability. However, it exhibits partial resilience to noise for widths less than σ_capprox 0.89, where the resilience order parameter takes values near 1 and then drops to zero at σ_c. We also use our results to shed light on the correctability of toric-rotor codes in higher dimensions d > 2. Our work shows that the quantum formalism for partition functions provides a mathematically rigorous framework for studying correctability in continuous-variable quantum codes.