Graph Relations Arising From Quantum Bernoulli Noises And Related Quantum Walks

Quantum walk(also known as quantum random walk)is the quantum analogue of classical random walk,which belongs to the research category of quantum stochas-tic analysis,and it has been intensively used in quantum information,quantum computation and mathematics-physics.Quantum Bernoulli noises are the family of annihilation and creation operators acting on Bernoulli functionals,which satisfy an anti-commutation relation(CAR)in equal-time,Because of the nice properties of annihilation and creation operators,quantum Bernoulli noises are applied in many problems of mathematics-physics,in this paper,we aim to discussing the applica-tion of quantum Bernoulli noises in quantum random walk.The main works are as follows:(1)We introduce a kind of graph relations associated with quantum Bernoulli noises(hereinafter referred to as " quantum Bernoulli noises graph")and investigate its properties.(2)We construct a model of continuous-time quantum walk on the quantum Bernoulli noises graph,then we examine its dynamic properties.(3)We introduce a model of discrete-time quantum walk on the quantum Bernoulli noises graph,and we examine the dynamic properties of the model and give the representation in the tensor space,especially,we show that it has the invariable probability distribution.