Universal Quantum Computation with Gapped Boundaries

TL;DR

This paper introduces a method for universal quantum computation using gapped boundaries in topological phases, providing systematic encoding, protected operations, and a concrete example with potential physical realization.

Contribution

It presents a new approach to topological quantum computation with gapped boundaries, including a universal gate set and a symmetry-protected charge measurement primitive.

Findings

Concrete example with $ ext{D}( ext{Z}_3)$ toric code

Potential realization in fractional quantum Hall systems

Universal quantum computation possible 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.