Using a biased quantum random walk as a quantum lumped element router

Quantum random walks have received much attention for their intrinsic interest and many possible uses and have been experimentally demonstrated. In this work we look at the possibility of using a biased one-dimensional (1D) quantum walk as an element within a larger quantum device. We ask whether one can use a quantum walk to act as a router with one bias setting engineering the quantum walk to route probability flow one direction while another bias setting routes flow in the opposite direction. Appealing to electrical circuit terminology, we consider a biased quantum walk over a large spatial lattice to act as a single “lumped element” whose routing action depends on the coin bias. We discover that the lumped-element current, when summed over the quantum walk lattice, reaches a steady state and for specific initial states we derive an analytic form for this steady-state lumped-element current. We show that we can control the magnitude and the direction (routing) of the steady-state current. Curiously the control phase and steady-state total current exhibits a sinusoidal current-phase relationship indicating that the lumped element may be similar to that found in Josephson junctions. Finally we illustrate that conservative 1D Hamiltonian systems can also exhibit steady-state dynamics similar to the quantum walk.

Figure 1

Using the “lumped-element” biased quantum walk as a router. We examine whether one can control the routing of an initial quantum state injected at the middle of a quantum walk to either end of the lumped element. We consider a 1D biased quantum walk with a large number of lattice sites as a lumped element, which can be used to redirect quantum probability to either end and into attached quantum wires (not described here). The bias parameters $(\theta ,\delta )$ parametrize the type of coin used in the quantum walk.

Figure 2

For a QW which is initialized to yield a spatially symmetric walk, where $({\alpha }_{n,0},{\beta }_{n,0})=\frac{1}{\sqrt{2}}(1,i){\delta }_{n,0}$, i.e., where the walker is initially located at the origin, and using $\theta =\pi /4,\delta =0$ (a Hadamard coin), we plot (top) the probability for the walker to be found at spatial location $n$ at time $t$, i.e., $\rho (n,t)$, and (bottom) the probability current density $j(n,t)$. We see that although the probability evolves in a spatially symmetric fashion the forward-difference defined probability current density breaks this symmetry.

Figure 3

Global interference term $Q\left(t\right)$ for a symmetric localized initial coin state $({\alpha }_{n,0},{\beta }_{n,0})=\frac{1}{\sqrt{2}}(1,i){\delta }_{n,0}$ and with coin parameters $\theta =\pi /4,\delta =5\pi /12$, smoothed in time using a moving window average of width 10. The solid line is the numerical result while the dashed line is the asymptotic value $Q_0$.

Figure 4

Total “steady” current $J_{\infty }$ against $\delta$ for various coin values $\theta$ for a symmetric localized initial coin state $({\alpha }_{n,0},{\beta }_{n,0})=\frac{1}{\sqrt{2}}(1,i){\delta }_{n,0}$. The points are the results of numerical simulation using Eq. (11) while the lines are the analytical result (20). We note that $J_{\infty }(-\delta )=-J_{\infty }\left(\delta \right)$; i.e., $J_{\infty }\left(\delta \right)$ is an odd function of $\delta$.

Figure 5

For a QW which is initialized in an identical fashion to Fig. 2, we plot (top) the probability density $\rho (n,t)$ and (bottom) the symmetric probability current density $j^C(n,t)$. We see that now the current density is symmetric in space.