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arXiv:2204.00060 · Physical Review A

Rogue waves in discrete-time quantum walks

A. R. C. Buarque, Wandearley S. Dias, F. A. B. F. de Moura, M. L. Lyra, G. M. A. Almeida

Published Jul 11, 2022• 12 citations
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Abstract

Rogue waves are rapid and unpredictable events of exceptional amplitude reported in various fields, such as oceanography and optics, with much of the interest being targeted towards their physical origins and likelihood of occurrence. Here, we use the all-round framework of discrete-time quantum walks to study the onset of those events due to a random phase modulation, unveiling its long-tailed statistics, survival time, and dependence upon the degree of randomness. We find the minimal disorder strength allowing for their occurrence to scale $\ensuremath{\propto}{N}^{\ensuremath{-}1/2}, N$ being the number of sites. Moreover, an extreme-value analysis converges to the Gumbel class of limiting distributions.

DOI
10.1103/physreva.106.012414

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Rogue waves are rapid and unpredictable events of exceptional amplitude reported in various fields, such as oceanography and optics, with much of the interest being targeted towards their physical origins and likelihood of occurrence. Here, we use the all-round framework of discrete-time quantum walks to study the onset of those events due to a random phase modulation, unveiling its long-tailed statistics, survival time, and dependence upon the degree of randomness. We find the minimal disorder strength allowing for their occurrence to scale \ensuremath∝N\ensuremath−1/2,N\ensuremath{\propto}{N}^{\ensuremath{-}1/2}, N\ensuremath∝N\ensuremath−1/2,N being the number of sites. Moreover, an extreme-value analysis converges to the Gumbel class of limiting distributions.
We investigate the emergence of rogue quantum amplitudes in discrete-time quantum walks (DTQWs) influenced by phase disorder. Our study reveals the statistics of occupation probability amplitudes in space and time, uncovering optimal disorder regimes that favor rogue wave events. Through numerical simulations, we demonstrate that the probability of rogue waves increases with quantum coins close to the Pauli-Z choice, regardless the disorder degree. Conversely, for coins near Pauli-X rogue events are scarce, except under weak disorder. A monotonic threshold is observed between rare- and high-probability rogue wave regimes, depending on the quantum coin. We provide a comprehensive analysis of the coin-disorder interplay to rogue wave events. Our findings shed light on the possible control of extreme quantum amplitudes through quantum coins in disordered DTQWs.
We investigate the emergence of rogue quantum amplitudes in discrete-time quantum walks (DTQWs) influenced by phase disorder. Our study reveals the statistics of occupation probability amplitudes in space and time, uncovering optimal disorder regimes that favor rogue wave events. Through numerical simulations, we demonstrate that the probability of rogue waves increases with quantum coins close to the Pauli-ZZZ choice, regardless of the disorder degree. Conversely, for coins near the Pauli-XXX choice, rogue events are scarce, except under weak disorder. A monotonic threshold is observed between rare and high-probability rogue wave regimes, depending on the quantum coin. We provide a comprehensive analysis of the coin-disorder interplay in regard to rogue wave events. Our findings shed light on the possible control of extreme quantum amplitudes through quantum coins in disordered DTQWs.
Extreme events are the ones for which the magnitude of the event is much larger than its typical values. They have been extensively investigated on classical dynamical and stochastic processes on networks. In this work, in analogy with well-studied extreme events of random walks, the extreme events of quantum walks on regular lattices, a scale-free network, and Erdős-Rényi network are studied. Due to unitary quantum evolution, the quantum walk dynamics usually differs from that of the classical random walks, and hence we expect extreme event properties to be different in classical and quantum settings. In contrast to this expectation, we report that extreme events of phase coherent quantum walks are qualitatively similar to that of classical random walks. We show that this counterintuitive behavior originates from the dominance of nodal fluctuations over internode correlations in disordered network structures. In particular, the occurrence probability for extreme events on scale-free and Erdős-Rényi networks displays a decaying power law with the degree of nodes. That is, the extreme event probability is larger for small degree nodes compared to the hubs on the network, in qualitative agreement with corresponding classical random walk results. Further, it is shown that extreme event probability scales with the threshold used to define extreme events.
We introduce a discrete-time quantum random walk (QRW) framework for spatial epidemic modelling on a two-dimensional square lattice and compare its dynamics to classical random-walk SIR models. In our model, each infected site spawns a quantum walker whose coherent evolution (controlled by an amplitude-splitting coin and conditional shifts) can infect visited susceptible sites with probability ppp and persists for a lifetime of τττ steps. We perform extensive quantum simulations on finite lattices and compute the basic reproduction number R0R_0R0​ across a broad grid of (p,τ)(p,τ)(p,τ) values. Results show that QRW dynamics interpolate between diffusive and super-diffusive regimes: at low ppp the QRW reproduces classical-like R0R_0R0​, while at higher ppp and τττ ballistic propagation and interference produce markedly larger R0R_0R0​ and non-Gaussian spatial profiles. We compare the QRW R0R_0R0​ range to empirical estimates from historical outbreaks and discuss parameter regimes where QRW offers a closer qualitative match than classical diffusion. We conclude that QRWs provide a flexible, conceptually novel toy model for exploring rapid or heavy-tailed epidemic spread.
We consider interacting many-particle systems with quenched disorder having strong Griffiths singularities, which are characterized by the dynamical exponent, zzz, such as random quantum systems and exclusion processes. In several d=1d=1d=1 and d=2d=2d=2 dimensional problems we have calculated the inverse time scales, ${\ensuremath{\tau}}^{\ensuremath{-}1}$, in finite samples of linear size, LLL, either exactly or numerically. In all cases, having a discrete symmetry, the distribution function, $P({\ensuremath{\tau}}^{\ensuremath{-}1},L)$, is found to depend on the variable, $u={\ensuremath{\tau}}^{\ensuremath{-}1}{L}^{z}$, and to be universal given by the limit distribution of extremes of independent and identically distributed random numbers. This finding is explained in the framework of a strong disorder renormalization group approach when, after fast degrees of freedom are decimated out, the system is transformed into a set of noninteracting localized excitations. The Fr\'echet distribution of $P({\ensuremath{\tau}}^{\ensuremath{-}1},L)$ is expected to hold for all random systems having a strong disorder fixed point, in which the Griffiths singularities are dominated by disorder fluctuations.
Quantum walks are powerful tools not only for constructing the quantum speedup algorithms but also for describing specific models in physical processes. Furthermore, the discrete time quantum walk has been experimentally realized in various setups. We apply the concept of the quantum walk to the problems in quantum foundations. We show that randomness and the arrow of time in the quantum walk gradually emerge by periodic projective measurements from the mathematically obtained limit distribution under the time-scale transformation.