Thermal stability of two-dimensional Topological Color Code

TL;DR

This paper investigates the thermal stability of the two-dimensional Topological Color Code, demonstrating that it is unstable at finite temperatures due to exponential decay of observables' auto-correlation functions, despite being stable at zero temperature.

Contribution

It provides a detailed analysis of the thermal decay processes in the Topological Color Code using Lindblad evolution and establishes a size-independent lower bound for decay rates.

Findings

Auto-correlation functions decay exponentially over time.

The code is unstable at finite temperature due to thermal fluctuations.

Decay rate lower bound is independent of system size.

Abstract

Thermal stability of the Topological Color Code in presence of a thermal bath is studied. We study the Lindblad evolution of the observables in the weak coupling limit of the Born-Markov approximation. The auto-correlation functions of the observables are used as a figure of merit for the thermal stability. We show that all of the observables auto-correlation functions decay exponentially in time. By finding a lower bound of the decay rate, which is a constant independent of the system size, we show that the Topological Color Code is unstable against thermal fluctuations from the bath at finite temperature, even though it is stable at T=0 against local quantum perturbations.