On the energy landscape of 3D spin Hamiltonians with topological order

TL;DR

This paper investigates the energy landscape of 3D spin Hamiltonians with topological order, demonstrating that certain models have energy barriers that grow logarithmically with system size, which is promising for quantum memory stability.

Contribution

It characterizes the energy landscape of 3D topologically ordered stabilizer Hamiltonians without string-like logical operators, proving a logarithmic lower bound on energy barriers for logical errors.

Findings

Energy barriers grow at least logarithmically with lattice size.

The logarithmic bound is tight for a specific 3D spin Hamiltonian.

Topologically ordered models without string operators can still have robust energy barriers.

Abstract

We explore feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have string-like logical operators. We prove that any sequence of local errors mapping a ground state of such Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is shown to be tight up to a constant factor for one particular 3D spin Hamiltonian.