ABSTRACT - DRUM - University of Maryland

modifies Ivanov’s rule of non-Abelian statistics. Since the bulk of the superconductor
is fully gapped such corrections are exponentially small. In the context of TQC such
deviations from Ivanov’s rule are sources of errors in single-qubit quantum gates.
Second, we consider dynamical transitions of Majorana fermions in the zeroenergy
ground states to excited states. The effects of such non-adiabatic transitions
strongly rely on whether these excited states are bound states with discrete spectrum
and localized at the same positions with the Majorana bound states, or they extend
through the whole bulk and form a continuum. Generally speaking, non-adiabatic
transitions mix the zero-energy ground states with other excited states and it is
questionable whether the quantum entanglement crucial to non-Abelian statistics is
still preserved. In the former case where excited states are localized, we are still able
to define conserved fermion parity stored in these low-energy bound states. Non-
Abelian statistics can be generalized once we enlarge the Hilbert space to include all
local bound states. In the latter case, the situation is dramatically different because
the notion of fermion parity in the low-energy Hilbert space no longer makes sense
once extended states above the bulk gap are involved.
We characterize the loss
of fermion parity in such non-adiabatic transitions by the expectation value of the
“local fermion parity” operator. This can be viewed as the dissipation of topological
qubit resulting from couplings to a continuum of fermionic states. We have thus
quantified the expectation that a zero-energy Majorana mode will decay if it is put
in contact with a continuum of fermionic states(e.g., electrons).
Although the underlying technological motivation for topological quantum
computation is that quantum error correction against continuous decoherence is un-
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