Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped Dirac equation

We study the dynamics of discrete-time quantum walk using quantum coin operations, $\widehat{C}({\theta }_1)$ and $\widehat{C}({\theta }_2)$ in time-dependent periodic sequence. For the two-period quantum walk with the parameters ${\theta }_1$ and ${\theta }_2$ in the coin operations we show that the standard deviation [${\sigma }_{{\theta }_1,{\theta }_2}(t)$] is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations ${\theta }_1$ or ${\theta }_2$, $\sigma_{\theta_1, \theta_2}(t) = \min {\sigma_{\theta_1}(t), \sigma_{\theta_2}(t) }$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walks, we present the bounds on the dynamics of three- and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using ${\theta }_1$ and ${\theta }_2$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\theta =0$, here we show the recovery of the massless Dirac equation with non-zero $\theta$ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.