Haar-random coin quantum walks on d-regular graphs yield a non-ergodic averaged channel that depolarizes the coin while preserving forever-measurable initial-state information in the vertex subspace for Cayley graphs of Abelian groups.
Quantum walks in higher dimensions
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abstract
We analyze the quantum walk in higher spatial dimensions and compare classical and quantum spreading as a function of time. Tensor products of Hadamard transformations and the discrete Fourier transform arise as natural extensions of the quantum coin toss in the one-dimensional walk simulation, and other illustrative transformations are also investigated. We find that entanglement between the dimensions serves to reduce the rate of spread of the quantum walk. The classical limit is obtained by introducing a random phase variable.
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Quantum random walks on d-regular graphs with Haar-random coin operators
Haar-random coin quantum walks on d-regular graphs yield a non-ergodic averaged channel that depolarizes the coin while preserving forever-measurable initial-state information in the vertex subspace for Cayley graphs of Abelian groups.