Riassunto analitico
In this Master Thesis we studied the evolution in time of Discrete-Time Quantum Walks on a one-dimensional lattice under many types of coins with different dimensions. For 2-dimensional coins the walker was able just to make one step forward or one step backward while for 3-dimensional coins it was able also to sit still. For 4-dimensional coins the walker could make instead a single or a double step either backward or forward. In particular we studied x,y and z Bloch rotations in 2,3 and 4 dimensions, embedded rotations in 3 and 4 dimensions obtained from lower-dimensional proper rotations and also the case of a 3-dimensional generic phase-diagonal coin. For the z and embedded rotations we managed to perform the computations analytically, as well as for some selected angles of rotation in the 2-dimensional x and y rotations. However, for the generic x and y 2-dimensional rotations and for the higher dimensional ones we had to rely on numerical simulations. The study of the 3-dimensional generic phase-diagonal coin was performed analytically too. In all the coins we left a free parameter (namely the angle of rotation) and in our study we mainly focused in finding the Quantum Fisher Information (QFI) related to that parameter and the initial states maximizing it. We also monitored the classical Fisher Information (FI) and other quantities related to the dispersion of the walker on the lattice, like the Shannon Entropy associated to the probability distribution in position of the walker and it's Inverse Population Ratio.
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