Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes

TL;DR

This paper explores the topological properties of Hamiltonian paths and their automorphisms, providing a framework for understanding automorphism codes like the honeycomb code in fault-tolerant quantum memory.

Contribution

It introduces a general framework for adiabatic Hamiltonian paths and constructs automorphism codes for topological orders, including a new perspective on the honeycomb code.

Findings

Constructed paths for automorphisms of 2D topological order

Connected adiabatic paths to automorphisms of the toric code

Provided a conjectured topological classification of Hamiltonian path spaces

Abstract

The recent "honeycomb code" is a fault-tolerant quantum memory defined by a sequence of checks which implements a nontrivial automorphism of the toric code. We argue that a general framework to understand this code is to consider continuous adiabatic paths of gapped Hamiltonians and we give a conjectured description of the fundamental group and second and third homotopy groups of this space in two spatial dimensions. A single cycle of such a path can implement some automorphism of the topological order of that Hamiltonian. We construct such paths for arbitrary automorphisms of two-dimensional doubled topological order. Then, realizing this in the case of the toric code, we turn this path back into a sequence of checks, constructing an automorphism code closely related to the honeycomb code.