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

• URL: http://arxiv.org/abs/2203.11137v2
• Date: Mon, 4 Apr 2022 16:51:19 GMT
• Title: Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes
• Authors: David Aasen, Zhenghan Wang, Matthew B. Hastings
• Abstract summary: "Honeycomb code" is a fault-tolerant quantum memory defined by a sequence of checks. We argue that a general framework to understand this code is to consider continuous adiabatic paths of gapped Hamiltonians. We turn this path back into a sequence of checks, constructing an automorphism code closely related to the honeycomb code.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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