Skeleton structure inherent in discrete-time quantum walks

In this paper we claim that a common underlying structure—a skeleton structure—is present behind discrete-time quantum walks (QWs) on a one-dimensional lattice with a homogeneous coin matrix. This skeleton structure is independent of the initial state, and partially, even of the coin matrix. This structure is best interpreted in the context of quantum-walk-replicating random walks (QWRWs), i.e., random walks that replicate the probability distribution of quantum walks, where this structure acts as a simplified formula for the transition probability. Additionally, we construct a random walk whose transition probabilities are defined by the skeleton structure and demonstrate that the resultant properties of the walkers are similar to both the original QWs and QWRWs.

Figure 1

Probability distributions ${\mu }_n\left(x\right)$ of QW or QWRW and functions $p_n\left(x\right)$ and $q_n\left(x\right)$ in the symmetric and asymmetric cases at time steps $n=500$. The parameters in both cases are summarized in Table 1. Each value is plotted only on even points; on odd points, the probabilities are 0. Functions $p_{500}\left(x\right)$ and $q_{500}\left(x\right)$ on the lower panel are expressed with a stacked bar graph, which is drawn only on the even points. Note that the relation $p_{500}\left(x\right)+q_{500}\left(x\right)=1$ holds for all $x$. The vertical dotted lines indicate the position of the peaks of ${\mu }_{500}$ at $x=\pm 500\left|a\right|$.

Figure 2

The relationship between $p_{500}\left(x\right)$ (blue line) and its skeleton structure $\tau \left(s\right)$ (red line) in each case.

Figure 3

The probability distribution of the QSRW at time $n=500$ [${\rho }_{500}\left(x\right)$, blue line] and comparison with the that of the corresponding QWRW [${\mu }_{500}\left(x\right)$, red line, dashed]. Each value is plotted only on the even points; on the odd points, the probabilities are 0.

Figure 4

Paths of walkers following (a) QSRW and (b) QWRW (the final time $n=500$, the number of walkers $K=100$). In both cases paths spread radially, which is caused by linear spreading.

Figure 5

Future direction following QSRW (red line; dashed) and QWRW (blue line) in the symmetric case on Table 1. The final time is $N=500$.

Figure 6

KL divergence between probability distributions between QWRW and QSRW in the symmetric case in Table 1, denoted by $D_{\mathrm{KL}}({\mu }_n\parallel {\rho }_n)$.

Figure 7

Graphs of $p_n\left(0\right)$ over time step $n$. Plots are given only to even $n$; if $n$ is odd, then $p_n\left(0\right)=1/2$ by the definition (31). The initial state $|\phi \rangle ={\left[10\right]}^{\mathsf{T}}$ for both cases.

Figure 8

Time variation of transition probabilities $p_n\left(x\right)$ and $q_n\left(x\right)$ in the case of Fig. 7. The values are expressed only on the even (resp. odd) points when the time step is even (resp. odd).

Figure 9

Functions $\psi (t,s)=sin\left(2t\theta \right(s\left)\right)$ and $\theta \left(s\right)$ ($t=20$).