Quantum Walks on Finite Graphs
In this chapter, we calculate the state of quantum walks on cycles, finite two-dimensional lattices, and hypercubes. The cycle is the finite version of the line. The finite lattice is the two-dimensional version of the cycle with the form of a torus. Finally, the hypercube is a generalization of the cube to dimensions greater than three. These graphs are basic ones but have interesting properties. They can be analyzed by means of the Fourier transform, allowing analytical calculations, which have many useful by-products in other contexts. In particular, the spectral decomposition of the quantum-walk evolution operator can be used in spatial search algorithms on these graphs. The spectral decomposition is described in details in this chapter.
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