This work will be devoted to the quantisation of the classical Bernoulli random walk over Z. As this random walk is isomorphic to the classical chaotic dynamical system x 7→ 2x mod 1 with x ∈ [0, 1], we will explore the rôle of classical periodic orbits of this chaotic map in relation with a non commutative algebra associated with the quantisation of the Bernoulli walk. In particular we show th...

There are presently two models for quantum walks on graphs. The “coined” walk uses discrete time steps, and contains, besides the particle making the walk, a second quantum system, the coin, that determines the direction in which the particle will move. The continuous walks operate with continuous time. Here a third model for quatum walks is proposed, which is based on an analogy to optical int...

This work will be devoted to the quantisation of the classical Bernoulli random walk over Z. As this random walk is isomorphic to the classical chaotic dynamical system x 7→ 2x mod 1 with x ∈ [0, 1], we will explore the rôle of classical periodic orbits of this chaotic map in relation with a non commutative algebra associated with the quantisation of the Bernoulli walk. In particular we show th...

The role of classical noise in quantum walks (QW) on integers is investigated in the form of discrete dichotomic random variable affecting its reshuffling matrix parametrized as a SU2)/U(1) coset element. Analysis in terms of quantum statistical moments and generating functions, derived by the completely positive trace preserving (CPTP) map governing evolution, reveals a pronounced eventual tra...

Recently, in [11], the “coin-turning walk” was introduced on Z. It is a non-Markovian process where steps form (possibly) time-inhomogeneous Markov chain. In this article, we follow up investigation by introducing analogous processes Zd,d≥2: at time n direction of “updated” with probability pn; otherwise next step repeats previous one. We study some fundamental properties these walks, such as t...