  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.
