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, $\hat{C}(θ_1)$ and $\hat{C}(θ_2)$ in time-dependent periodic sequence. For the two-period quantum walk with the parameters $θ_1$ and $θ_2$ in the coin operations we show that the standard deviation [$σ_{θ_1, θ_2} (t)$] is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations $θ_1$ or $θ_2$, $σ_{θ_1, θ_2}(t) = \min \{σ_{θ_1}(t), σ_{θ_2}(t) \}$. 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 $θ_1$ and $θ_2$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $θ=0$, here we show the recovery of the massless Dirac equation with non-zero $θ$ 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.

10 Pages, 9 figures, Published version