Abstract:
Quantum simulation become a necessary step to learn physics about a system when theoretical analysis, direct experimental observation, and numerical investigations in classical computers are difficult. The Dirac particles in a general situation is one such system. Discrete quantum walk (DQW)---a U(2) coin operations followed by a coin state dependent positional shift operations is a powerful quantum simulation scheme, and implementable in well controllable table-top set-ups. In the thesis, we first identify that the conventional DQW can't exactly simulate Dirac Cellular Automaton (DCA) which is a discretized theory of free Dirac Hamiltonian (DH). We found some particular choice of coin parameters of the split-step (SS) DQW---a generalization of DQW can fully simulate single-particle DCA. Next we question whether the same SS-DQW can simulate dynamics of free Dirac particle with extra degrees of freedom like colors, flavors besides the spin or chirality. One such example is Neutrino oscillation. By moving from the U(2) coined SS-DQW to the U(6) coined SS-DQW we have simulated the exact probability profile of Neutrino flavor transitions. We further probe towards simulating single particle massive DH in presence of background potentials and space-time curvature. By using a SS-DQW with position-time dependent coin parameters, and we realize that it will give us an unbounded effective Hamiltonian, at the continuum limit of position-time. So we have introduced a modified version of inhomogeneous SS-DQW which will produce a bounded effective Hamiltonian. This modified SS-DQW with U(2) coin operations produces single-particle massive DH in presence of abelian gauge potentials and space-time curvature. Introducing higher dimensional---U(N) coin operations in the modified SS-DQW we can include non-abelian potentials in the same DH. In order to simulate two-particle DH in presence of curved space-time and external potentials, we have used two particle modified SS-DQW, where the shift operations act separately on each particle, the coin operations which act simultaneously on both particles and contain all kinds of interactions.
