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arXiv:1909.08527

Diverse tunable dynamics of two quantum random walkers

Shrabanti Dhar, Khaleque, Abdul, Tushar Kanti Bose

Published Sep 17, 2019• 3 citations
arXivHTMLPDF

Abstract

Quantum walk research has mainly focused on evolutions due to repeated applications of time-independent unitary coin operators. However, the idea of controlling the single particle evolution using time-dependent unitary coins has still been a subject of multiple studies as it not only hosts interesting possibilities for quantum information processing but also opens a much richer array of phenomena including static and dynamic localizations. So far, such studies have been performed only for single quantum walkers. In case of multi-walker systems, time-dependent coins may generate measurable phenomena not described by the single-particle model, due to entanglement and interaction among the walkers. In this context, we present here a thorough numerical study of an one dimensional system of two quantum walkers exhibiting rich collective dynamics controlled by simple time-dependent unitary coins proposed in [Phys. Rev. A \textbf{80}, 042332(2009)] and [Phys. Rev. A \textbf{73},062304(2006)]. We study how the interplay of coin time-dependence, simple interaction schemes, entanglement and the relative phase between the coin states of the particles influences the evolution of the quantum walk. The results show that the system offers a rich variety of collective dynamical behavior while being controlled by time dependent coins. In particular, we find and characterize fascinating two-body localization phenomena with tunable quasiperiodic dynamics of correlations and entanglements which are quantities of quantum origin.

DOI
10.48550/arxiv.1909.08527

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Quantum walk research has mainly focused on evolutions due to repeated applications of time-independent unitary coin operators. However, the idea of controlling the single particle evolution using time-dependent unitary coins has still been a subject of multiple studies as it not only hosts interesting possibilities for quantum information processing but also opens a much richer array of phenomena including static and dynamic localizations. So far, such studies have been performed only for single quantum walkers. In case of multi-walker systems, time-dependent coins may generate measurable phenomena not described by the single-particle model, due to entanglement and interaction among the walkers. In this context, we present here a thorough numerical study of an one dimensional system of two quantum walkers exhibiting rich collective dynamics controlled by simple time-dependent unitary coins proposed in [Phys. Rev. A \textbf{80}, 042332(2009)] and [Phys. Rev. A \textbf{73},062304(2006)]. We study how the interplay of coin time-dependence, simple interaction schemes, entanglement and the relative phase between the coin states of the particles influences the evolution of the quantum walk. The results show that the system offers a rich variety of collective dynamical behavior while being controlled by time dependent coins. In particular, we find and characterize fascinating two-body localization phenomena with tunable quasiperiodic dynamics of correlations and entanglements which are quantities of quantum origin.
Abstract Quantum walks are known to propagate quadratically faster than their classical counterparts and are used to model dynamics in various quantum systems. The spread of the quantum walk in position space shows anomalous diffusion behavior. By controlling the action of quantum coin operation on the corresponding coin degree of freedom of the walker, one can demonstrate control over the diffusion behavior. In this work, we report different forms of coin operations on quantum walks exhibiting anomalous diffusion behavior. Homogeneous and accelerated quantum walks display superdiffusive behavior, whereas uncorrelated static and dynamic disorders in the evolution induce strong and weak localization of the particle indicating subdiffusive and normal diffusive behavior. The role played by the interference effects in the spreading of the walker has remained elusive and our aim in this work is to present the interplay between quantum coherence and mean squared displacement of the walker. We employ two reliable measures of coherence for conclusively establishing the role of quantum interference as the driving force behind the anomalous diffusive behavior in the dynamics of quantum walks.
Quantum magic, which accounts for the nonstabilizer content of a state, is essential for universal quantum computation beyond classically simulable resources. However, the way magic builds up during structured unitary dynamics remains largely open. Here, we investigate the generation and evolution of quantum magic in discrete-time quantum walks (DTQWs)—a simple, tunable unitary model realizable on a wide range of architectures. We use stabilizer Rényi entropy as a measure of quantum magic and investigate single- and two-walker quantum walks on a one-dimensional lattice, considering a wide range of initial coin states. Our results reveal that DTQWs can dynamically generate significant magic, with the amount and structure dependent on the initial state of the coin. In the case of a single walker, we found a nontrivial and complementary relationship between magic and entanglement at long times. In the two-walker setting, even states close to stabilizer form can evolve into highly magical states when subjected to quantum walk protocols. The generation of magic can occur independently of entanglement growth, emphasizing its distinct role as a quantum resource. We further show that the dynamical magic generation is robust under realistic noise—specifically, decoherence in the coin degree of freedom—demonstrating that this process persists across noisy settings. Finally, we discuss an experimentally feasible scheme to measure magic using current technology. Our findings position DTQWs as accessible and controllable platforms for producing quantum magic, offering a fresh perspective on their role in quantum information processing and reliable quantum computation.
We report on the possibility of controlling quantum random walks (QWs) with a step-dependent coin (SDC). The coin is characterized by a (single) rotation angle. Considering different rotation angles, one can find diverse probability distributions for this walk including: complete localization, Gaussian and asymmetric likes. In addition, we explore the entropy of walk in two contexts; for probability density distributions over position space and walker's internal degrees of freedom space (coin space). We show that entropy of position space can decrease for a SDC with the step-number, quite in contrast to a walk with step-independent coin (SIC). For entropy of coin space, a damped oscillation is found for walk with SIC while for a SDC case, the behavior of entropy depends on rotation angle. In general, we demonstrate that quantum walks with simple initiatives may exhibit a quite complex and varying behavior if SDCs are applied. This provides the possibility of controlling QW with a SDC.
We offer theoretical explanations for some recent observations in numerical simulations of quantum random walks (QRWs). Specifically, in the case of a QRW on the line with one particle (walker) and two entangled coins, we explain the phenomenon, called ``localization,'' whereby the probability distribution of the walker's position is seen to exhibit a persistent major ``spike'' (or ``peak'') at the initial position and two other minor spikes which drift to infinity in either direction. Another interesting finding in connection with QRWs of this sort pertains to the limiting behavior of the position probability distribution. It is seen that the probability of finding the walker at any given location eventually becomes stationary and nonvanishing. We explain these observations in terms of the degeneration of some eigenvalue of the time evolution operator U(k)U(k)U(k). An explicit general formula is derived for the limiting probability, from which we deduce the limiting value of the height of the observed spike at the origin. We show that the limiting probability decreases quadratically for large values of the position xxx. We locate the two minor spikes and demonstrate that their positions are determined by the phases of nondegenerated eigenvalues of U(k)U(k)U(k). Finally, for fixed time ttt sufficiently large, we examine the dependence on ttt of the probability of finding a particle at a given location xxx.
We investigate a discrete-time quantum walk of a three-component quantum particle on a one-dimensional lattice. As coin operators, we employ parameterized rotations generated by the Gell-Mann matrices, which enable systematic tuning of the couplings between the internal components. We analyze how these couplings influence and control the dynamics by systematically exploring the position-space probability distribution across a broad region of the parameter space. To quantify the impact of different inter-component couplings, we further examine the ratio of the mean position to the variance in each half of the lattice. Our results indicates that the system supports a rich variety of transport regimes, ranging from nearly symmetric, rapidly spreading walks to strongly anisotropic dynamics with partial localization. This framework thus provides a new avenue for engineering targeted spreading and trapping behavior in multicomponent discrete-time quantum walks.
Abstract We investigate the evolution of a discrete-time one-dimensional quantum walk driven by a position-dependent coin. The rotation angle, which depends upon the position of a quantum particle, parameterizes the coin operator. For different values of the rotation angle, we observe that such a coin leads to a variety of probability distributions, e.g. localized, periodic, classical-like, semi-classical-like, and quantum-like. Further, we study the Shannon entropy associated with position and the coin space of a quantum particle, and compare them with the case of the position-independent coin. Our results show that the entropy is smaller for most values of the rotation angle as compared to the case of the position-independent coin. We also study the effect of entanglement on the behavior of probability distribution and Shannon entropy by considering a quantum walk with two identical position-dependent entangled coins. We observe that in general, a wave function becomes more localized as compared to the case of the position-independent coin and hence the corresponding Shannon entropy is lower. Our results show that a position-dependent coin can be used as a controlling tool of quantum walks.