One-Dimensional And Multi-Dimensional Quantum Walk And Physical Implementation

Classical random walk has been widely used in many fields.Quantum random walk-a quantum counterpart of classical walk,has different features with classical walk.Since it was put forward in 1993,it has become a hot topic for more and more researchers.Quantum walk spread quadratically faster than classical walk.Thus quantum walk would be more efficient in the development of the algorithm,and can be used to build a general quantum computing model.Nowadays,quantum walk is a trustable way to the research of quantum computation and is full of exciting open problems for physicists,computer scientists and engineers.In order to play the role of quantum walk better,we made a more indepth study of the quantum walk.In this paper,we first discussed the characteristics of one dimensional quantum walk with quadratic position-dependent phase disorder.Under certain conditions,the walker will return to the initial position within a fixed number of steps.In addition,the degree of attenuation is related to the phase modulation parameters.With the increase of the phase modulation parameters,the probability of walker returning back to the initial position is increased.The probability distribution is periodic,and the period varies with the changing of phase modulation parameters.When the number of steps is large enough,the probability distribution shows the trend of oscillatory attenuation.Secondly,we propose a scheme to realize two dimensional quantum walk.One dimensional quantum walk has been realized in many systems.There are only very few schemes to achieve two dimensional quantum walk.However,a large number of quantum processes must be realized in two-dimensional or higher dimensional quantum walk.In this paper,we use linear optical elements to design a reasonable and effective scheme to realize the universal two-dimensional quantum walk.And a method of expanding the dimension of the Hilbert space of coins is introduced.By increasing the spatial modes of photons,the purpose of expanding the Hilbert space of coins is achieved.And the number of experimental elements required in our scheme increase linearly with the growth of steps.Thus our scheme can be used to implement a large number of steps of multi-dimensional quantum walk feasibility and stability.