Effective quantum memory Hamiltonian from local two-body interactions

TL;DR

This paper demonstrates how a self-correcting quantum memory model with long-range interactions can be realized using only local two-body interactions, by deriving an effective Hamiltonian from a Heisenberg ferromagnet and employing perturbative gadgets.

Contribution

It shows that the key ingredients for a self-correcting quantum memory can be implemented with only two-body interactions, simplifying physical realization.

Findings

Effective Hamiltonian derived from a Heisenberg ferromagnet

Perturbative gadgets generate five-spin operators

Expected self-correcting properties under low-energy conditions

Abstract

In [Phys. Rev. A 88, 062313 (2013)] we proposed and studied a model for a self-correcting quantum memory in which the energetic cost for introducing a defect in the memory grows without bounds as a function of system size. This positive behavior is due to attractive long-range interactions mediated by a bosonic field to which the memory is coupled. The crucial ingredients for the implementation of such a memory are the physical realization of the bosonic field as well as local five-body interactions between the stabilizer operators of the memory and the bosonic field. Here, we show that both of these ingredients appear in a low-energy effective theory of a Hamiltonian that involves only two-body interactions between neighboring spins. In particular, we consider the low-energy, long-wavelength excitations of an ordered Heisenberg ferromagnet (magnons) as a realization of the bosonic field. Furthermore, we present perturbative gadgets for generating the required five-spin operators. Our Hamiltonian involving only local two-body interactions is thus expected to exhibit self-correcting properties as long as the noise affecting it is in the regime where the effective low-energy description remains valid.