Topological Quantum Computation with Gapped Boundaries

TL;DR

This paper explores fault-tolerant quantum computation using gapped boundaries in topological models, classifying excitations, describing boundary defects, and demonstrating universal quantum gates with surface codes.

Contribution

It introduces a comprehensive framework for gapped boundaries in quantum double models, including classification, Hamiltonian realization, and applications to universal quantum computation.

Findings

Classified boundary excitations in Kitaev's models.

Provided Hamiltonians for boundary defects.

Demonstrated universal quantum gates with surface codes.

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.