# Phase transition in a noisy Kitaev toric code model

## TL;DR

This paper explores how classical phase transitions in the 2D Ising model influence the quantum properties of the noisy Kitaev toric code, revealing a transition from coherence to decoherence linked to classical criticality.

## Contribution

It establishes a duality between classical phase transitions and quantum decoherence in the toric code, introducing an order parameter that mirrors the Ising magnetization and analyzing stability at critical noise levels.

## Key findings

- Quantum decoherence transition corresponds to classical phase transition.
- Order parameter behaves like Ising magnetization, indicating robust coherence.
- Toric code exhibits increased stability at the classical critical point.

## Abstract

It is well-known that the partition function of a classical spin model can be mapped to a quantum entangled state where some properties on one side can be used to find new properties on the other side. However, the consequences of the existence of a classical (critical) phase transition on the corresponding quantum state has been mostly ignored. This is particularly interesting since the classical partition function exhibits non-analytic behavior at the critical point and such behavior could have important consequences on the quantum side. In this paper, we consider this problem for an important example of Kitaev toric code model which has been shown to correspond to the two-dimensional (2D) Ising model though a duality transformation. Through such duality transformation, it is shown 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 non-coherent 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 decoherency of such a mixture and show that it behaves similar to the order parameter (magnetization) of 2D Ising model, a behavior that is interpreted as a robust coherency 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. We close the paper with a discussion on connection between the robust coherency as well as the critical stability with topological order of the toric code model.