Many aspects of the well-known mapping between the partition function of a classical spin model and the quantum entangled state have been studied in recent years. However, the consequences of the existence of a classical (critical) phase transition on the corresponding quantum state have been mostly ignored. In this paper, we consider this problem for an important example of the Kitaev toric code model which has been shown to correspond to the two-dimensional (2D) Ising model though a duality transformation. We show that the temperature on the classical side is mapped to bit-flip noise on the quantum side. It is then shown that a transition from a coherent superposition of a given quantum state to a noncoherent mixture corresponds exactly to paramagnetic-ferromagnetic phase transition in the Ising model. To identify such a transition further, we define an order parameter to characterize the decoherence of such a mixture and show that it behaves similar to the order parameter (magnetization) of the 2D Ising model, a behavior that is interpreted as a robust coherence in the toric code model. Furthermore, we consider other properties of the noisy toric code model exactly at the critical point. We show that there is a relative stability to noise for the toric code state at the critical noise which is revealed by a relative reduction in susceptibility to noise.

• Received 23 December 2018

DOI:https://doi.org/10.1103/PhysRevA.99.052312