Discrete-time quantum walk approach to state transfer

We show that a quantum-state transfer, previously studied as a continuous-time process in networks of interacting spins, can be achieved within the model of discrete-time quantum walks with a position-dependent coin. We argue that, due to additional degrees of freedom, discrete-time quantum walks allow one to observe effects which cannot be observed in the corresponding continuous-time case. First, we study a discrete-time version of the engineered coupling protocol due to Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and then we discuss the general idea of conversion between continuous-time quantum walks and discrete-time quantum walks.

Figure 1

Because the interference phenomenon occurs for the second-nearest neighbors, even vertices can be interpreted as edges.

Figure 2

A DTQW on a finite chain is effectively simulated by a walk on a cycle. The coin operator associated to one of the vertices corresponding to edges (see Fig. 1) flips the coin state and as a result reverses the walk.

Figure 3

The evolution of perfect-state transfer from position $1$ to position $N-1$ for DTQW with position-dependent coin ($N=30$ and $\lambda =0.03$). Thin (blue) line, probability of occupying position 1; thick (purple) line, probability of occupying position $N-1$; dashed line, probability of occupying remaining vertices of the chain.

Figure 4

Arguments $\phi \in [-\pi ,\pi ]$ of eigenvalues $e^{i\phi }$ of a position-dependent coin DTQW double-step operator $U^2$ for $N=30$ and $\lambda =0.03$. Each dot corresponds to a doubly degenerated eigenvalue.

Figure 5

Eigenstate population for $N=30$, $\theta =\pi /4$, and initial state $|1,\to \rangle$: (top) ${\theta }_1={\theta }_{N-1}=\theta =\pi /4$; (bottom) ${\theta }_2={\theta }_{N-2}=\pi /2-\varepsilon$, where $\varepsilon =\frac{\pi }{2N}$. The initial state is almost entirely supported on four eigenstates only.

Figure 6

Probability of population of positions $1$ (solid line) and $N-1$ (dashed line) in subsequent time steps: $N=30$, $\theta =\frac{\pi }{4}$, and $\varepsilon =\frac{\pi }{2N}$.