M.Sc Thesis
M.Sc Student Streltsin Vladislav Braiding Majoranas away From the Adiabatic Regime Department of Physics Professor Netanel Lindner

Abstract

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.