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 h be the space of square integrable complex-valued Bernoulli functionals, which is infinitely dimensional. First, we construct on a given subspace h_L ⊂ h a self-adjoint operator Δ_L via the canonical unitary involutions on h, and by analyzing its spectral structure we find out all its eigenvalues. Then, we introduce an abstract model of CTQW with h_L as its state space, which is governed by the Schr\"{o}dinger equation with Δ_L 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 Δ_L as well as to the model itself. Finally, we prove that the model has PST at time t=π/2. Some other interesting results are also proven of the model.