Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems

TL;DR

This paper investigates the thermal fragility of topological quantum order in surface code models, showing that topological symmetry operators lose expectation value at any non-zero temperature, affecting quantum information protection.

Contribution

It introduces the concept of thermal fragility in topologically ordered systems and provides explicit autocorrelation functions for Kitaev's model, highlighting the impact of finite temperature.

Findings

Topological symmetry operators vanish at any non-zero temperature.

Autocorrelation times can be large at low temperatures.

Proliferation of topological defects causes loss of correlations.

Abstract

Are systems that display Topological Quantum Order (TQO), and have a gap to excitations, hardware fault-tolerant at finite temperatures? We show that in surface code models that display low d-dimensional Gauge-Like Symmetries, such as Kitaev's and its generalizations, the expectation value of topological symmetry operators vanishes at any non-zero temperature, a phenomenon that we coined thermal fragility. The autocorrelation time for the non-local topological quantities in these systems may remain finite even in the thermodynamic limit. We provide explicit expressions for the autocorrelation functions in Kitaev's model. If temperatures far below the gap may be achieved then these autocorrelation times, albeit finite, can be made large. The physical engine behind the loss of correlations at large spatial and/or temporal distance is the proliferation of topological defects at any finite temperature as a result of a dimensional reduction. This raises an important question: How may we best quantify the degree of protection of quantum information in a topologically ordered system at finite temperature?