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A new model maps a quantum random walk described by a Hadamard operator to aparticular case of a birth and death process. The model is represented by a 2DMarkov chain with a stochastic matrix, i.e., all the transition rates arepositive, although the Hadamard operator contains negative entries (this ispossible by increasing the dimensionality of the system). The probabilitydistribution of the walker population is preserved using the Markovianproperty. By applying a proper transformation to the population distribution ofthe random walk, the probability distributions of the quantum states |0>, 1>are revealed. Thus, the new model has two unique properties: it reveals theprobability distribution of the quantum states as a unitary system andpreserves the population distribution of the random walker as a Markoviansystem.

Mapping a Hadamard Quantum Walk to a Unique Case of a Birth and Death Process