ABSTRACT - DRUM - University of Maryland

three different ways: (a) tunneling of non-Abelian anyons when there are multiples
of them, which splits the degenerate ground state manifold and therefore introduces
additional dynamical phases in the evolution; (b) transitions to excited bound states
outside the Hilbert space of zero-energy states, in this case topologically protected
braidings have to be defined within an enlarged Hilbert space; (c) transitions to
the continuum of extended states which render the fermion parity in the low-energy
Hilbert space ill-defined. These non-adiabatic effects are possible sources of errors
for quantum gates in TQC. The main goal of this chapter is to quantitatively address
these effects and their implications on quantum computation.
Our work is the first systematic attempt to study non-adiabaticity in the
anyonic braiding of non-Abelian quantum systems. Given that the braiding of non-
Abelian anyons is the unitary gate operation [136] in topological quantum computation
[100], understanding the dynamics of braiding as developed in this work
is one of the keys to understanding possible errors in topological quantum computation.
The other possible source of error in topological quantum computation
is the lifting of the ground state anyonic degeneracy due to inter-anyon tunneling,
which we have studied elsewhere [91, 106]. Although we study the braiding nonadiabaticity
in the specific context of the topological chiral p-wave superconductors
using the dynamical BdG equatons within the BCS theory, our work should be of
general validity to all known topological quantum computation platforms, since all
currently known non-Abelian anyonic platforms in nature are based on the SU(2) 2
conformal field theory of Ising anyons, which are all isomorphic to the chiral p-
wave topological superconductors [100].
As such, our work, with perhaps some
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