Infinite reduction in absorbing time in quantum walks over classical ones

Abstract

We study the absorption time and spreading rate of the discrete-time quantum walk propagating on a line in the presence or absence of an absorber. We analytically establish that in the presence of an absorber, the average absorption time of the quantum walker is finite, contrary to the behavior of a classical random walker, indicating an infinite resource reduction on moving over to a quantum version of a walker. Furthermore, numerical simulations indicate a reversal of this behavior due to the insertion of disorder in the walker's step lengths. Additionally, we demonstrate that in the presence of an absorber, there is a speed-up in the spreading rate, and that a disordered quantum walk that is sub-ballistic regains the ballistic spreading of a clean quantum walk.

Figures (6)

• Figure 1: A comparison of the disorder-averaged absorbing time of CRW and DTQW, both with the presence of an absorber at the position $2$. The Poisson distribution with unit mean is the underlying disorder. The red line, without any marker, corresponds to the disordered-CRW. This plot shows a sudden fall at the beginning and no significant change is observed in the $\langle t_{a} ^{(n)}\rangle$ values after few initial steps. As $n$ increases, $\langle t_{a} ^{(n)}\rangle$ attends a constant value of $7.75$ (approximately), showing the convergence nature of $(\langle t_{a} ^{(n)}\rangle)_n$ for the classical walk. Here, we have considered numerical simulations up to $n=400$ steps. The black, asterisk-marked plot shows the behavior of the disorder-averaged absorbing time of disordered-DTQW defined by the Hadamard coin. This plot exhibits the monotonically increasing nature of $(\langle t_{a} ^{(n)}\rangle)_n,$ within the considered range of $n$ values up to $400$ steps ($\langle t_{a} ^{(n)}\rangle\approx 38$ at $n=400$). This signifies that $(\langle t_{a} ^{(n)}\rangle)_n$ diverges for the DTQW we considered. Here, the averaged time quantity is derived over $40$ different disorder realization runs.
• Figure 2: Probability distribution of 1D-DTQW. This position-probability plot describes the probability distribution nature of the 1D-DTQW after $50$ time steps, starts with the initial state $\ket{0}\otimes \ket{0}.$ Here, the black vertical lines quantify the probability of the walker being at the specified positions on the line, when there is no absorber on the line. Here some higher probability values are observed along the positive nodes (right-hand side) compared to the other side. On the other hand, the red colour represents the case when there is an absorber at position $2.$ As a result of the absorber present at node $2$, most of the non-zero probabilities are noticed along the negative nodes, with the highest peak around $-32.$
• Figure 3: Plot of $\ln(\sigma)$ against $\ln(t)$. The linear fitting (on the log-log plot) for the data points (blue colored dots) corresponding to the Hadamard 1D-DTQW with one absorber placed at $2$ is represented by red solid line. With $95\%$ confidence level, the slope of the fitted line falls within the confidence intervals $0.96\pm 0.005$, with an average least square error of $0.002$ for the linear fittings. To ensure fewer errors in the fitting, we have chosen the $t$-range $20 < t < 80$.
• Figure 4: Diagram of probability distribution of 1D Poisson disordered-DTQW defined by Hadamard coin after $50$ evolutions, for a single disorder realization with mean value $1$, without and with the presence of the absorber. In both cases, the walker starts from $0$ node with initial coin state $\ket{0}$, and the outcomes of the Poisson distribution of unit mean in a single run are taken as the disorder realizations. The black color lines show probability distributions of disordered-DTQW over different locations, without the presence of any absorber. The red bars represent the position-probability distribution of disordered-DTQW, where an absorber is kept at node $2.$ In contrast to the no absorber case, non-zero probabilities are seen to be more localized near the starting position $0$ when the absorber is placed.
• Figure 5: (Color online.) $\ln(\sigma)$ against $\ln(t)$ for disordered (Poisson distribution) CRWs with and without absorber scenarios. Here, $t$ ranges between $20$ and $80.$ The red and blue fitted lines represent the cases without and with an absorber, respectively. Red line equation is $\ln(\sigma) = 0.51 \ln(t) + 0.31,$ and the blue line is $\ln(\sigma) = 0.51 \ln(t) -0.14$. The slopes of both lines are the same.