Discrete Time
A quantum walk in discrete time is specified by a coin and shift operator, which are applied repeatedly.
Consider what happens when we discretize a massive Dirac operator over one spatial dimension. In the absence of a mass term, we have left-movers and right-movers. They can be characterized by an internal degree of freedom, "spin" or a "coin". When we turn on a mass term, this corresponds to a rotation in this internal "coin" space. A quantum walk corresponds to iterating the shift and coin operators repeatedly.
This is very much like Feynman's model of an electron in 1 spatial and 1 time dimension. He summed up the zigzagging paths, with left-moving segments corresponding to one spin (or coin), and right-moving segments to the other. See Feynman checkerboard for more details.
Read more about this topic: Quantum Walk
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