ABSTRACT - DRUM - University of Maryland

In the case of Ising anyons, 2N vortices can realize N −1 qubits. A topological
qubit thus requires at least 4 vortices. Let us label the four Majorana zero modes
as γ i , i = 1, 2, 3, 4. We can construct two complex fermionic modes c 1 = 1 √
2
(γ 1 +
iγ 2 ), c 2 = 1 √
2
(γ 3 + iγ 4 ) and the degenerate states can be specified by the occupation
numbers of c 1 and c 2 . Fixing the global fermion parity to be even, the two qubit
states can be specified by the occupation numbers in the fermionic states |00〉 and
|11〉 = c † 1c † 2|00〉. Braiding of the vortices generates π 2 rotations e±i π 4 σx,y,z = 1 √
2
(1 ±
iσ x,y,z ). The braids can be used to generate other single-qubit rotations, such as a
Hardamard gate H:
H = 1 √
2
(σ x + σ z ) = 1 √
2
(1 + iσ y )σ z . (1.25)
So H can be implemented as a NOT gate (braiding twice) followed by another braid.
We can go on and consider two qubits constructed from six vortices and braidings
generate two-qubit gates in addition to single-qubit operations.
In addition, one also needs to find ways to read out the quantum information
stored in the topological qubits. Let us still take the Ising anyons as an example.
Since the qubit states are labeled by the fermion parity eigenvalues in pairs of
vortices, to read out the qubit is amount to measure the fermion parity contained in
a finite spatial region, which can be done typically by interferometry experiments [38,
39]. The basic idea is to exploit the fact that in a superconductor, when a fermion
goes around a superconducting vortex, a π Berry phase is experienced by the fermion
due to Aharonov-Bohm effect [40]. We have already made crucial use of this fact
when deriving the non-Abelian statistics of vortices in p x +ip y superconductors. Now
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