Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low temperature regime

TL;DR

This paper investigates the relaxation dynamics of finite-size topological qubits in contact with a thermal bath, revealing a finite-size and temperature-dependent crossover behavior that impacts the stability of topological quantum memories.

Contribution

It introduces a finite-size scaling analysis of the toric code's relaxation time at low temperatures, highlighting a size-dependent crossover temperature and its implications.

Findings

Finite-size scaling of relaxation time governed by classical random walks.

Identification of a size-dependent crossover temperature inversely proportional to log(system size).

Stronger finite-temperature effects on stability below the crossover temperature.

Abstract

We present an analysis of the relaxation dynamics of finite-size topological qubits in contact with a thermal bath. Using a continuous-time Monte Carlo method, we explicitly compute the low-temperature nonequilibrium dynamics of the toric code on finite lattices. In contrast to the size-independent bound predicted for the toric code in the thermodynamic limit, we identify a low-temperature regime on finite lattices below a size-dependent crossover temperature with nontrivial finite-size and temperature scaling of the relaxation time. We demonstrate how this nontrivial finite-size scaling is governed by the scaling of topologically nontrivial two-dimensional classical random walks. The transition out of this low-temperature regime defines a dynamical finite-size crossover temperature that scales inversely with the log of the system size, in agreement with a crossover temperature defined from equilibrium properties. We find that both the finite-size and finite-temperature scaling are stronger in the low-temperature regime than above the crossover temperature. Since this finite-temperature scaling competes with the scaling of the robustness to unitary perturbations, this analysis may elucidate the scaling of memory lifetimes of possible physical realizations of topological qubits.