Recurrence in discrete-time quantum stochastic walks

TL;DR

The paper investigates recurrence in a discrete-time quantum stochastic walk on a line, revealing that adding classical randomness can nontrivially suppress recurrence in certain parameter regimes. It introduces a monitored-recurrence framework and first-return generating functions to analyze the asymptotic return probabilities across the quantum–classical interpolation defined by p. A key finding is the existence of a threshold θ_* ≈ 0.289π where R(p) develops a dip for small p, indicating robust quantum–classical interplay. The study also contrasts two DTQSW constructions, showing that coin-state decoherence can enforce classical recurrence (R=1 for p>0) and discussing experimental routes to observe these effects. Overall, the work highlights that stochasticity can yield algorithmic advantages in discrete-time quantum walks and provides a rigorous methodology for assessing recurrence in open quantum dynamics.

Abstract

Interplay between quantum interference and classical randomness can enhance performance of various quantum information tasks. In the present paper we analyze recurrence phenomena in the discrete-time quantum stochastic walk on a line, which is a quantum stochastic process that interpolates between quantum and classical walk dynamics. Surprisingly, we find that introducing classical randomness can reduce the recurrence probability -- despite the fact that the classical random walk returns with certainty -- and we identify the conditions under which this intriguing phenomenon occurs. Numerical evaluation of the first-return generating function allows us to investigate the asymptotics of the return probability as the step number approaches infinity. This provides strong evidence that the suppression of recurrence probability is not a transient effect but a robust feature of the underlying quantum-classical interplay in the asymptotic limit. Our results show that for certain tasks discrete-time quantum stochastic walks outperform both classical random walks and unitary quantum walks.

Figures (10)

• Figure 1: Estimate of recurrence probability evaluated with the generating function (\ref{['rec:prob:z']}) for different values of the coin angle $\theta$ as a function of the parameter $p$. We used $z=0.99999$, comparable to $t_{\mathrm{eff}} = 10^5$ steps.
• Figure 2: Recurrence probability for $\theta_*$ (blue), $\theta_* - 0.1\,\text{rad}$ (orange) and $\theta_* + 0.1\,\text{rad}$ (green), where the estimate $0.2892\pi$ was substituted for the unknown exact value. In (a) we show the return probability after 5 steps (lower lines in the legend) and 10 steps (higher three lines). For 5 steps all three curves are decreasing as functions of $p$. At 10 steps, all return probability values increase, with the change more pronounced with higher $p$, and likewise do the derivatives at $p = 0$ of all three displayed cases. The 10-step return probability becomes an increasing function for $\theta=\theta_*-0.1$, while the other two cases maintain a negative slope near $p = 0$. (b) shows the final recurrence probability estimated using the generating functions with $z=0.99999$. One can see that for $\theta>\theta_*$ the initial decrease survives while the rightmost value approaches 1, forming a dip.
• Figure 3: Numerically computed first derivative of $R_t$ at zero $R_t'(p=0) = B_t$ against the coin angle $\theta$ with focus on the parameter range where $B_t$ crosses zero, with $t$ chosen to grow in increasing steps. The simulation indicates that $\theta_* \gtrsim 0.28915 \pi$.
• Figure 4: Recurrence probability of the DTQSW where we interpolate between a QW and the correlated random which has the same coin angle $\theta$. We have chosen $\theta = \frac{2\pi}{5}$. Blue line is obtained for $t=5$ steps, orange dashed curve corresponds to the $t=10$ steps and green dotted curve to $t=100$ steps. For the first 5 steps the return probability is independent of $p$. After 5 steps the dependence on $p$ emerges, however, $R_t(p)$ is always an increasing function. Notice that with increasing number of steps $t$ the slope at $p=0$ becomes steeper. The red dot-dashed curve is evaluated using the generating functions. The data correspond to the choice $z=0.99999$, i.e. $t_\text{eff} = 10^5$. Asymptotically, the leftmost point is pinned by \ref{['surv:p=0']}, which for the used $\theta$ predicts $R \approx 0.9224$, but all points at $p > 0$ tend to 1.
• Figure 5: Fit results for varying $z$ parameter. The fitting function is $a - b(1-z)^c$. The best fit value and standard error of the exponent $c$ are plotted for various combinations of $\theta$ and $p$. The following values of $z$ were sampled: 0.99, 0.995, 0.998, 0.999, 0.9995, 0.9998, 0.9999, 0.99995, 0.99998, 0.99999, and $N_{\text{max}}$ was set to 20.