Masters Defense: Magnus Fosse Bøe – University of Copenhagen

Masters Defense: Magnus Fosse Bøe

Majorana surface codes with dislocations

The planar code is at present the most promising proposal for developing fault-tolerant
quantum computation. Building on the investigations of twist defects in the toric code
model [1, 2], this thesis considers the possibility of constructing computational qubits
by introducing dislocations in the planar code.
We review the basic properties of Majorana Fermions (MFs) and the toric code.
By use of Schrie er-Wol perturbation theory we find that the planar code model with
twist defects emerges as the low-energy sector of an array of tunnel coupled Majorana-
Cooper boxes (MCBs), with the twist defects being unpaired MFs. We fi nd that the
system under investigation can be initialized, and that the MFs can be moved around
in the array. By introduction of a modi ed, split MCB, we are able to perform braiding
operations with the MFs. We calculate the Berry phase of the braiding process in two
diff erent ways, and confi rm that it agrees with the expected result.
Finally, we describe how the state of the system can be measured. This is required
in order to observe the non-Abelian statistics of these particles. The operations we can
carry out with this system are not enough for universal quantum computation, but in
combination with other techniques, such as the T-gate (a /8-rotation), implementable
through for example magic state destillation [3, 4], universality can be achieved.