Feasibility of self-correcting quantum memory and thermal stability of topological order

TL;DR

This paper investigates the thermal stability of topologically ordered systems and the feasibility of self-correcting quantum memory, concluding that such stability cannot be achieved in 2D and 3D systems with certain Hamiltonian models.

Contribution

It introduces a quantum code model based on stabilizer Hamiltonians and demonstrates its limitations as a self-correcting quantum memory due to topological constraints.

Findings

The model cannot serve as self-correcting quantum memory.

Systems with this model cannot have thermally stable topological order.

Thermal stability and local perturbation resistance cannot coexist in these systems.

Abstract

Recently, it has become apparent that the thermal stability of topologically ordered systems at finite temperature, as discussed in condensed matter physics, can be studied by addressing the feasibility of self-correcting quantum memory, as discussed in quantum information science. Here, with this correspondence in mind, we propose a model of quantum codes that may cover a large class of physically realizable quantum memory. The model is supported by a certain class of gapped spin Hamiltonians, called stabilizer Hamiltonians, with translation symmetries and a small number of ground states that does not grow with the system size. We show that the model does not work as self-correcting quantum memory due to a certain topological constraint on geometric shapes of its logical operators. This quantum coding theoretical result implies that systems covered or approximated by the model cannot have thermally stable topological order, meaning that no system can be stable against both thermal fluctuations and local perturbations simultaneously in two and three spatial dimensions.