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

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.

36 pages, 10 figures; v2: additional references and minor revisions

References in corpus (8)

• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Dynamically Generated Logical Qubits
• A Fault-Tolerant Honeycomb Memory
• Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
• Higher-dimensional generalizations of the Berry curvature
• Boundaries for the Honeycomb Code
• Mapping between Morita equivalent string-net states with a constant depth quantum circuit
• Phases and phase transitions of a perturbed Kekulé-Kitaev model

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• Anyon condensation and the color code
• Topology, criticality, and dynamically generated qubits in a stochastic measurement-only Kitaev model
• Pauli topological subsystem codes from Abelian anyon theories
• Crystalline Quantum Circuits
• Higher Berry Phase of Fermions and Index Theorem
• Gauging Lie group symmetry in (2+1)d topological phases
• On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking