M.Sc Student | Streltsin Vladislav |
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Subject | Braiding Majoranas away From the Adiabatic Regime |

Department | Department of Physics |

Supervisor | Professor Netanel Lindner |

Full Thesis text |

Physical systems
which realize certain topological phases present the interesting possibility of
storing and manipulating quantum information in a manner which is protected
from noise and decoherence. Notable examples are topological superconductors,
which possess boundary zero modes known as “Majorana fermion zero modes”.
Exchange of two Majorana zero modes (MZMs) is called “braiding” since it yields
a (projective) representation of the braid group. Therefore, braiding of MZMs
can be utilized to manipulate the quantum state of the system in a
topologically robust manner. In principle, the braiding process is defined in
the adiabatic limit where it is done infinitely slowly with respect to the
timescale defined by the gap of the system. In this work we examine the
braiding protocol when it is performed within a finite time. We find that
finite braiding times imply imperfect braiding results that strongly depend on
the details of the process. We will demonstrate that in Hamiltonian dynamics
there is no way to find “optimal ways” to carry out the braiding process and
there are infinitely many ways to “fine tune” the details of the process, in
order to get results which are arbitrarily close to the ideal (adiabatic) ones.
In dephasing Lindbladian dynamics, on the other hand, we will see that one can,
in fact, formulate an optimization problem. We will show that non-adiabaticity
implies two types of errors: “out-of-subspace” errors which cause spurious
excitations of the system, and “phase errors” which alter the transformation
within the ground state manifold. We will show that “out-of-subspace” errors
scale with some power of the adiabatic parameter *e = 1/Delta*T*, where *T*
is the finite time of the protocol and *Delta* is the gap of the system.
We show that the leading order contribution can be controlled by smoothness of
the Hamiltonian at the endpoints. In addition, we show that “out-of-subspace”
errors and “phase errors” vanish simultaneously. Finally, we show how to
eliminate these errors completely using measurements at specific times of the
braiding process.