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

Abstract

We study the dynamics of discrete-time quantum walk using quantum coin operations, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}({\ensuremath{\theta}}_{1})$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}({\ensuremath{\theta}}_{2})$, in time-dependent periodic sequence. For the two-period quantum walk with the parameters ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$ in the coin operations we show that the standard deviation $[{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1},{\ensuremath{\theta}}_{2}}(t)]$ is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations ${\ensuremath{\theta}}_{1}$ or ${\ensuremath{\theta}}_{2}$, ${\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1},{\ensuremath{\theta}}_{2}}(t)=\mathrm{min}{{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{1}}(t),{\ensuremath{\sigma}}_{{\ensuremath{\theta}}_{2}}(t)}$. 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 ${\ensuremath{\theta}}_{1}$ and ${\ensuremath{\theta}}_{2}$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\ensuremath{\theta}=0$, here we show the recovery of the massless Dirac equation with nonzero $\ensuremath{\theta}$ 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.