The discrete time quantum walk was defined as the quantum mechanical analogy of the random walk. Like the random walk, there will be expected to be many applications to various fields, especially in quantum information science. Also, this was experimentally demonstrated in various type physical systems. In this presentation, we have shown that the discrete time quantum walk can be taken as the quantum dynamical simulator in the one-dimensional system for asymptotic behaviors. It is emphasized that the non-relativistic quantum mechanical dynamics with the discrete time and lattice can be mapped to the relativistic quantum mechanical dynamics.

[1] Y. Shikano, "Time in Weak Value and Discrete Time Quantum Walk - From Quantum Measurement to Quantum Dynamics" (LAP LAMBERT Academic Publishing, Germany, 2012).