A profound consequence of certain classes of topological quantum states
of matter is the possibility of topologically protected degeneracies in
the ground state spectrum of the system. No local operators can
distinguish topologically degenerate states, allowing them to be
particularly robust to decoherence. In this talk, I will discuss some
new directions in the pursuit of creating, measuring, and manipulating
topological degeneracies, which utilize the physics of gapped
boundaries. Some specific results I will discuss include (1) a proposal
for creating and reading out topological states without localized zero
modes in bilayer Laughlin fractional quantum Hall states; and (2) ideas
for how to implement topologically robust unitary transformations when
the topological degeneracy is due to disconnected gapped boundaries and
there are no localized zero modes.