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

We propose and study a model of a quantum memory that features self-correcting properties and a lifetime growing arbitrarily with system size at nonzero temperature. This is achieved by locally coupling a two-dimensional $L\times L$ toric code to a three-dimensional (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 $\ln \left(L\right)$. 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.

Figure 1

A 2D toric code [blue (dark) area in $xy$ plane] of size $L\times L$ is centered inside a cubic lattice of size ${\Lambda }^3$ with $\Lambda >L$. The stabilizers $W_p$ of the toric code locally couple to a system of hopping bosons on a cubic lattice. A long-range attraction between the stabilizers is mediated by the low-energy collective excitations of the bosons.

Figure 2

Two tilings of plaquettes are shown on which the code may be defined. Spins are located on vertices. (a) The square tiling, as usually employed for the toric code. $s$ plaquettes are shown in dark blue (black), blue (gray), or light blue (light gray); $p$ plaquettes are shown in white. (b) An alternative tiling, with alternating triangular and octagonal plaquettes. $s$ plaquettes are shown in dark blue (black) and blue (gray); $p$ plaquettes are shown in white and gray (light gray).

Figure 3

Numerical evaluation of the sum ${\sum }_p\frac{1}{|{\mathbf{R}}_p|}$ as function of $L$ for a lattice constant $a=1$. The sum increases linearly with $L$, in agreement with the continuum approximation calculation.