Universal Quantum Computation with Gapped Boundaries

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 $mathfrak{D}(mathbb{Z}₃), is discussed. For this example, we have a qutrit encoding and an abstract universal gate set. Physically, gapped boundaries of\mathfrak{D}mathbb{Z}₃can be realized in bilayer fractional quantum Hall1/3$ 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.