Quantum walk in terms of quantum Bernoulli noises

Quantum Bernoulli noises are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we first present some new results concerning quantum Bernoulli noises, which themselves are interesting. Then, based on these new results, we construct a time-dependent quantum walk with infinitely many degrees of freedom. We prove that the walk has a unitary representation and hence belongs to the category of the so-called unitary quantum walks. We examine its distribution property at the vacuum initial state and some other initial states and find that it has the same limit distribution as the classical random walk, which contrasts sharply with the case of the usual quantum walks with finite degrees of freedom.