Modular transformations through sequences of topological charge projections

Abstract

The ground state subspace of a topological phase of matter forms a representation of the mapping class group of the space on which the state is defined. We show that elements of the mapping class group of a surface of genus $g$ can be obtained through a sequence of topological charge projections along at least three mutually intersecting non-contractible cycles. We demonstrate this both through the algebraic theory of anyons and also through an analysis of the topology of the space-time manifold. We combine this result with two observations: (i) that surfaces of genus $g$ can be effectively simulated in planar geometries by using bilayer, or doubled, versions of the topological phase of interest, and inducing the appropriate types of gapped boundaries; and (ii) that the required topological charge projections can be implemented as adiabatic unitary transformations by locally tuning microscopic parameters of the system, such as the energy gap. These observations suggest a possible path towards effectively implementing modular transformations in physical systems. In particular, they also show how the $\text{Ising}\otimes \overline{\text{Ising}}$ state, in the presence of disconnected gapped boundaries, can support universal topological quantum computation.

21 pages. v2: Added appendix describing protocol for realizing the pi/8 phase gate in the Ising x Ising* state with gapped boundaries and topological charge measurements. v3: updated author affiliation

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• Quantum Origami: Transversal Gates for Quantum Computation and Measurement of Topological Order
• Optical Lattice with Torus Topology
• Topological wormholes
• Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes
• Quantum Statistics and Spacetime Topology: Quantum Surgery Formulas
• Instantaneous braids and Dehn twists in topologically ordered states
• A comparative study of universal quantum computing models: towards a physical unification
• Fractional Chern insulator edges and layer-resolved lattice contacts
• Exactly soluble model of boundary degeneracy
• Measuring Topological Order
• On Topological Quantum Computing With Mapping Class Group Representations
• A graph-based formalism for surface codes and twists
• Fractional quantum Hall states with gapped boundaries in an extreme lattice limit