Topological Quantum Computation with Gapped Boundaries

Abstract

This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protected operations and obtain quantum gates. To demonstrate a potential physical realization, we provide quantum circuits for surface codes that can perform all basic operations on gapped boundaries. Finally, we show how gapped boundaries of the abelian theory $\mathfrak{D}({\mathbb{Z}}_3)$ can be used to perform universal quantum computation.

117 pages, 91 figures; minor changes from v1, added references, corrected typos

References in corpus (30)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Surface codes: Towards practical large-scale quantum computation
• Measurement-based quantum computation with cluster states
• String-net condensation: A physical mechanism for topological phases
• State preservation by repetitive error detection in a superconducting quantum circuit
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Detecting arbitrary quantum errors via stabilizer measurements on a sublattice of the surface code
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Quantum codes on a lattice with boundary
• Condensate induced transitions between topologically ordered phases
• Genons, twist defects, and projective non-Abelian braiding statistics
• Topological Nematic States and Non-Abelian Lattice Dislocations
• Topological boundary conditions in abelian Chern-Simons theory
• The Quantum Double Model with Boundary: Condensations and Symmetries
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Theory of defects in Abelian topological states
• Correspondences of ribbon categories
• Diagrammatics for Bose condensation in anyon theories
• Boson Condensation in Topologically Ordered Quantum Liquids
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Ground-state degeneracy for abelian anyons in the presence of gapped boundaries
• Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers
• Bogomolov multiplier, double class-preserving automorphisms and modular invariants for orbifolds
• Fermion Condensation and Gapped Domain Walls in Topological Orders
• Kitaev models based on unitary quantum groupoids
• Charge $2e/3$ superconductivity and topological degeneracies without localized zero modes in bilayer fractional quantum Hall states
• On Enriching the Levin-Wen model with Symmetry
• Nested Topological Order
• Exactly soluble model of boundary degeneracy

Cited by in corpus (16)

• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Universal Quantum Computation with Gapped Boundaries
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes
• String Condensations in 3+1D and Lagrangian Algebras
• Characterizing topological order by the information convex
• Instantaneous braids and Dehn twists in topologically ordered states
• Universal quantum computation in the surface code using non-Abelian islands
• Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits
• Galois Orbits of TQFTs: Symmetries and Unitarity
• Extend The Levin-Wen Model To Two-dimensional Topological Orders With Gapped Boundary Junctions
• Bulk-to-boundary anyon fusion from microscopic models
• Beyond Anyons
• Gauging Defects in Quantum Spin Systems
• Fractional quantum Hall states with gapped boundaries in an extreme lattice limit