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Here, a new two-dimensional process, discrete in time and space, that yieldsthe results of both a random walk and a quantum random walk, is introduced.This model describes the population distribution of four coin states |1>,-|1>,|0> -|0> in space without interference, instead of two coin states |1>, |0>.For the case of no boundary conditions, the model is similar to a Markov chainwith a stochastic matrix, i.e., it conserves the population distribution of thefour coin states, and by using a proper transformation, yield probabilitydistributions of the two quantum states |1>, |0> in space, similar to a unitaryoperator. Numerical results for a quantum random walk on infinite and finitelines are introduced.

Mapping quantum random walks onto a Markov chain by mapping a unitary transformation to a higher dimension of an irreducible matrix