Self-correcting quantum memory with a boundary

TL;DR

This paper investigates how boundary conditions affect the lifetime of quantum information in a 2D toric code with long-range interactions, revealing that boundaries can significantly influence system stability and lifetime growth.

Contribution

It demonstrates that boundary conditions impact the lifetime of quantum memory in long-range interacting toric codes, with detailed analysis of boundary effects and limitations.

Findings

Lifetime grows exponentially with system size for both boundary types

Boundary effects can be beneficial or detrimental to quantum memory stability

Upper bounds on system size due to perturbative treatment limitations

Abstract

We study the two-dimensional toric code Hamiltonian with effective long-range interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow to increase the lifetime of the stored quantum information arbitrarily by making $L$, the linear size of the memory, larger [Phys. Rev. A 82 022305 (2010)]. We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough $L$. This influence can be both beneficial or detrimental. In particular, we study an effective Hamiltonian proposed in [Phys. Rev. B 83 115415 (2011)] that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in $L^2$ for both periodic and open boundary conditions, though the exponent in the latter case is found to be less favourable. However, $L$ is upper-bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.