Abstract model of continuous-time quantum walk based on Bernoulli functionals and perfect state transfer

Abstract

In this paper, we present an abstract model of continuous-time quantum walk (CTQW) based on Bernoulli functionals and show that the model has perfect state transfer (PST), among others. Let [Formula: see text] be the space of square integrable complex-valued Bernoulli functionals, which is infinitely dimensional. First, we construct on a given subspace [Formula: see text] a self-adjoint operator [Formula: see text] via the canonical unitary involutions on [Formula: see text], and by analyzing its spectral structure we find out all its eigenvalues. Then, we introduce an abstract model of CTQW with [Formula: see text] as its state space, which is governed by the Schrödinger equation with [Formula: see text] as the Hamiltonian. We define the time-average probability distribution of the model, obtain an explicit expression of the distribution, and, especially, we find the distribution admits a symmetry property. We also justify the model by offering a graph-theoretic interpretation to the operator [Formula: see text] as well as to the model itself. Finally, we prove that the model has PST at time [Formula: see text]. Some other interesting results about the model are also proved.