Enhanced thermal stability of the toric code through coupling to a bosonic bath

TL;DR

This paper introduces a model coupling a 2D toric code to a 3D bosonic bath, enhancing its thermal stability and self-correcting properties, with energy barriers scaling favorably with system size.

Contribution

It provides an exactly solvable model demonstrating how coupling to a bosonic bath can significantly increase the energy barrier for anyon creation, improving quantum memory stability.

Findings

Energy penalty for anyon creation grows linearly with system size when coupled via displacement operator.

Energy penalty scales with logarithm of system size when coupled via density operator.

Quantum memory lifetime can grow exponentially or polynomially with system size depending on the coupling scheme.

Abstract

We propose and study a model of a quantum memory that features self-correcting properties and a lifetime growing arbitrarily with system size at non-zero temperature. This is achieved by locally coupling a 2D L x L toric code to a 3D bath of bosons hopping on a cubic lattice. When the stabilizer operators of the toric code are coupled to the displacement operator of the bosons, we solve the model exactly via a polaron transformation and show that the energy penalty to create anyons grows linearly with L. When the stabilizer operators of the toric code are coupled to the bosonic density operator, we use perturbation theory to show that the energy penalty for anyons scales with ln(L). For a given error model, these energy penalties lead to a lifetime of the stored quantum information growing respectively exponentially and polynomially with L. Furthermore, we show how to choose an appropriate coupling scheme in order to hinder the hopping of anyons (and not only their creation) with energy barriers that are of the same order as the anyon creation gaps. We argue that a toric code coupled to a 3D Heisenberg ferromagnet realizes our model in its low-energy sector. Finally, we discuss the delicate issue of the stability of topological order in the presence of perturbations. While we do not derive a rigorous proof of topological order, we present heuristic arguments suggesting that topological order remains intact when perturbative operators acting on the toric code spins are coupled to the bosonic environment.