Coassociative grammar, periodic orbits, and quantum random walk over ℤ

Motivated by the success of classical random walks and chaotic dynamical systems, we study the quantisation of the random walk over Z and its relationships with a classical chaotic system x → 2xmod1, x ∈ [0,1]. In the physics literature, quantum random walks have been studied, for instance, by Ambainis et al. [1] and Konno et al. [8, 9]. In [8], Konno shows that the quantum random walk over Z called the Hadamard random walk generates a particular combinatorics. In [6], Joni and Rota showed that some combinatorics can be recovered from coproducts of coassociative coalgebras. Therefore, is it possible to create a coassociative coalgebra which recovers the combinatorics generated by the Hadamard random walk? We start in Section 2 with briefly recalling a new formalism, inspired by weighted directed graph theory. In Section 3, we present a mathematical framework for studying the Hadamard random walk over Z. In Section 4, we construct a coassociative coalgebra based on results on graphs developed in Section 2. We show that the combinatorics generated by the Hadamard random walk over Z can be recovered from this coalgebra. In Section 5, we present the notion of quantum graphs developed in [15] and point out a relation between a quantum graph, the classical Bernoulli random walk, the Hadamard random walk, and the periodic orbits of the classical chaotic system x → 2xmod1 with x ∈ [0,1]. Briefly speaking, we show how to relate periodic orbits of this classical chaotic system to polynomials describing a quantisation of the Bernoulli walk.