Abstract Model of Continuous-Time Quantum Walk Based on Bernoulli Functionals and Perfect State Transfer

TL;DR

This paper introduces an abstract model of continuous-time quantum walks based on Bernoulli functionals, demonstrating perfect state transfer and analyzing spectral properties within an infinite-dimensional space.

Contribution

It develops a novel quantum walk model using Bernoulli functionals and proves its capability for perfect state transfer at a specific time.

Findings

Model exhibits perfect state transfer at t=π/2

Spectral analysis reveals all eigenvalues of the operator

Provides a graph-theoretic interpretation of the model

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