Coined Walks with Cyclic Boundary Conditions
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In this chapter, we address coined quantum walks on three important finite graphs: cycles, finite two-dimensional lattices , and hypercubes: A cycle is a finite version of the line; a finite two-dimensional lattice is a two-dimensional version of the cycle in the form of a discrete torus; and a hypercube is a generalization of the cube to dimensions greater than three. These graphs have spatial symmetries and can be analyzed via the Fourier transform method. We obtain analytic results that are useful in other chapters of this book. For instance, here we describe the spectral decomposition of the quantum walk evolution operators of two-dimensional lattices and hypercubes. The results are used in Chap. 9 in the analysis of the time complexity of spatial search algorithms using coined quantum walks on these graphs.