Coupling Of Hadamard Walk And Quantum Walk Based On Quantum Bernoulli Noise

As a quantum analogue of classical random walk,quantum walk has important applications in many fields.A one-dimensional two-state quantum walk is discrete-time quantum walk on one-dimensional integer lattice determined by a 2 × 2 uni-tary matrix,Hadamard walk its typical form.Quantum walk based on quantum Bernoulli noise is a discrete-time quantum walk with infinitely many internal de-grees of freedom on a one-dimensional integer lattice,which has strong decoherence.The purpose of this paper is to study the coupling and evolution of Hadamard walk and QBN walk,the main works are as follows.Firstly,we construct a new family coin operators on the coin space of the QBN walk by the coin operators of the Hadamard walk and QBN walk,and get a quantum walk model on two-dimensional integer lattice,which can be called the coupling walk of the Hadamard walk and QBN walk.The state space of coupling walk is l2(Z2,H),where H is a square integrable Bernoulli functional space.We obtain the evolution operator of coupling walk and its unitary representation.Secondly,by using the canonical isomorphism,we get the expression of the couping walk evolution operator in the tensor space l2(Z2)(?)H,and then give the corresponding form of the coupling walk evolution equation in the tensor space.Thirdly,by examining the Fourier representation of the coupling walk,we fur-ther discuss its construction properties.In addition,we also obtain the limit proba-bility distribution of the coupling model,and find the limit probability distribution of the coupling walk is the product of the Konno distribution and the Gaussian dis-tribution.This property indicates that the evolutionary behavior of coupling walk has both ballistic properties and diffusion properties,evolving at speed n and(?)in the horizontal and vertical directions respectively.