Invariance in quantum walks with time-dependent coin operators

This paper unveils some features of a discrete-time quantum walk on the line whose coin depends on the temporal variable. After considering the most general form of the unitary coin operator, it focuses on the role played by the two phase factors that one can incorporate there, and shows how both terms influence the evolution of the system. A closer analysis reveals that the probabilistic properties of the motion of the walker remain unaltered when the update rule of these phases is chosen adequately. This invariance is based on a symmetry with consequences not yet fully explored.

Figure 1

Probability mass function of the process after $t=100000$ time steps. The red solid line connects the points obtained by direct application of the evolution operator on the initial state when we pick at random (a) ${\alpha }_t$ and (b) ${\beta }_t$. Only probabilities corresponding to even values of $n$ are represented, as odd values have probability equal to zero. The blue dotted line corresponds to the limiting Gaussian probability density function.

Figure 2

Comparison of the wave function after $t=30$ time steps. The red solid lines and dots correspond to a time-homogeneous QW. The blue dotted lines show the real parts of the magnitudes associated with a time-dependent QW, while the imaginary parts are depicted by green dashed lines.