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Coined discrete-time quantum walks are studied using simple deterministicdynamical systems as coins whose classical limit can range from beingintegrable to chaotic. It is shown that a Loschmidt echo like fidelity plays acentral role and when the coin is chaotic this is approximately thecharacteristic function of a classical random walker. Thus the classicalbinomial distribution arises as a limit of the quantum walk and the walkerexhibits diffusive growth before eventually becoming ballistic. The coin-walkerentanglement growth is shown to be logarithmic in time as in the case ofmany-body localization and coupled kicked rotors, and saturates to a value thatdepends on the relative coin and walker space dimensions. In a coin dominatedscenario, the chaos can thermalize the quantum walk to typical random statessuch that the entanglement saturates at the Haar averaged Page value, unlike ina walker dominated case when atypical states seem to be produced.

Quantum walks with quantum chaotic coins: Of the Loschmidt echo, classical limit and thermalization