Limiting Distribution and Mixing Time
In this chapter, we use the notion of quantum walks on finite regular graphs with the goal of analyzing the limiting probability distribution and the mixing time. In finite quantum systems, there is a quasi-periodic pattern in the time evolution, preventing the convergence to a limiting distribution. The quasi-periodic behavior of the quantum state can be obtained from the expression of the eigenvalues of the evolution operator. A possible way to obtain limiting configurations is to define a new distribution called average probability distribution, which evolves stochastically and does not have the quasi-periodic behavior.
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