Universal Quantum Computation with Gapped Boundaries

Abstract

This Letter discusses topological quantum computation with gapped boundaries of two-dimensional topological phases. Systematic methods are presented to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, we introduce a new and general computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the Letter, a concrete physical example, the ${\mathbb{Z}}_3$ toric code ($\mathfrak{D}({\mathbb{Z}}_3)$), is discussed. For this example, we have a qutrit encoding and an abstract universal gate set. Physically, gapped boundaries of $\mathfrak{D}({\mathbb{Z}}_3)$ can be realized in bilayer fractional quantum Hall $1/3$ systems. If a practical implementation is found for the required topological charge measurement, these boundaries will give rise to a direct physical realization of a universal quantum computer based on a purely abelian topological phase.

4+4 pages, 25 figures, adapted from Ch. 5.1-4, 6.3 of arXiv:1609.02037; v2: very close to published version

References in corpus (15)

• Fault-tolerant quantum computation by anyons
• Surface codes: Towards practical large-scale quantum computation
• String-net condensation: A physical mechanism for topological phases
• Models for gapped boundaries and domain walls
• Quantum codes on a lattice with boundary
• Fractional topological insulators
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Classification and analysis of two dimensional abelian fractional topological insulators
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Universal Quantum Computation with Metaplectic Anyons
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Measurability of Wilson loop operators
• Topological Quantum Computation with Gapped Boundaries
• Charge $2e/3$ superconductivity and topological degeneracies without localized zero modes in bilayer fractional quantum Hall states
• Modular transformations through sequences of topological charge projections

Cited by in corpus (33)

• Prediction of Toric Code Topological Order from Rydberg Blockade
• Topological and subsystem codes on low-degree graphs with flag qubits
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• Dualities in one-dimensional quantum lattice models: topological sectors
• Anyon condensation and the color code
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• The boundaries and twist defects of the color code and their applications to topological quantum computation
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes
• Symmetry-protected self-correcting quantum memories
• Quantum neural networks to simulate many-body quantum systems
• Entanglement bootstrap approach for gapped domain walls
• Computing data for Levin-Wen with defects
• Robustness of Topological Order in the Toric Code with Open Boundaries
• Fusing Binary Interface Defects in Topological Phases: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ case
• Domain walls in topological phases and the Brauer-Picard ring for $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$
• Qutrit Toric Code and Parafermions in Trapped Ions
• Numerical investigation of gapped edge states in fractional quantum Hall-superconductor heterostructures
• Attractive Haldane bilayers for trapping non-Abelian anyons
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Computing Defects Associated to Bounded Domain Wall Structures: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ Case
• Beyond Anyons
• A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• Gauging Defects in Quantum Spin Systems
• Matrix formulation for non-Abelian families
• Fractional quantum Hall states with gapped boundaries in an extreme lattice limit
• Braiding defects in topological stabiliser codes of any dimension cannot be universal
• Topological Quantum Computation with Gapped Boundaries and Boundary Defects
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• Classification of Gapped Domain Walls of Topological Orders in 2+1 dimensions: A Levin-Wen Model Realization
• Measuring the Unique Identifiers of Topological Order Based on Boundary-Bulk Duality and Anyon Condensation
• Universal fault tolerant quantum computation in 2D without getting tied in knots