We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limits. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability
p∼1/nβ can be evaluated, where
n is the final time and
0<β<1. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter
β shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time
n. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.
A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A family of two-dimensional walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these discrete-time quantum walks (DTQWs) exhibit an exact discrete local U(1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called $\mathbf{E}\ifmmode\times\else\texttimes\fi{}\mathbf{B}$ drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.