ABSTRACT - DRUM - University of Maryland

equation (6.40), we find
ˆγ →
(1 − 4β2 Et
sin2
E2 2
ˆξ → 2√ 2εβ sin 2 Et
2
ˆη →
E 2
√
2β sin Et
ˆγ −
E
ˆγ+
)ˆγ+ 2√ 2βε sin 2 Et
2
E 2
(
cos Et + 4β2 sin 2 Et
2
E 2
ε sin Et
E
ˆξ + cos Et ˆη
√
2β sin Et
ˆξ− ˆη
E
ε sin Et
)ˆξ+
E ˆη
(6.42)
which should be followed up by the basis transformation ˆB. Here we have defined
E = √ ε 2 + 4β 2 .
By using (6.24) we can work out explicitly how the ground state wavefunctions
transform. Physically, the non-adiabatic process causes transitions of quasiparticles
residing on the zero-energy level to the excited levels. Superficially these transitions
to excited states significantly affect the non-Abelian statistics, since the parity of
fermion occupation in the ground state subspaces is changed as well as the quantum
entanglement between various ground states [20, 146].
This can also be directly
seen from (6.42) since starting from |g〉 the final state is a superposition of |g〉 and
ˆd † ˆd † 0 λ
|g〉. So we might suspect that errors are introduced to the gate operations.
However, noticing that the excited states are still localized, they are always
transported together with the zero-energy Majorana states. Threfore the parity of
the total fermion occupation in the ground state subspace and local excited states are
well conserved. This observation allows for a redefinition of the Majorana operators
to properly account for the fermion occupation in local excited states, as being done
in [111]. We therefore have to represent the fermion parity in the following way:
∏
ˆP 12 = −iˆγ 1 ˆξ1ˆη 1ˆγ 2 ˆξ2ˆη 2 = iˆγ 1ˆγ 2 (1 − 2 ˆd † ˆd i i ) (6.43)
shared by defects 1 and 2. Accordingly, we define the generalized Majorana operators
118
i=1,2