We study the dynamics of discrete-time quantum walk using quantum coin operations, $\widehat{C}\left({\theta }_1\right)$ and $\widehat{C}\left({\theta }_2\right)$, 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 $\left[{\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)\right]$ is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations ${\theta }_1$ or ${\theta }_2$, ${\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)=\min \{{\sigma }_{{\theta }_1}\left(t\right),{\sigma }_{{\theta }_2}\left(t\right)\}$. 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 nonzero $\theta$ parameters contributing to the intriguing interference in the dynamics in a totally nonrelativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.

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• Quantum algorithms
• Quantum entanglement
• Quantum simulation
• Quantum walks