Modular transformations through sequences of topological charge projections

TL;DR

This paper demonstrates how sequences of topological charge projections can realize the entire mapping class group of a surface, enabling effective implementation of modular transformations and universal topological quantum computation.

Contribution

It establishes a method to generate mapping class group elements via topological charge projections and connects this to practical implementation in physical systems.

Findings

Sequences of topological charge projections realize mapping class group elements.

Surface genus can be simulated in planar geometries with gapped boundaries.

Topological charge projections can be implemented as adiabatic unitary transformations.

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.