Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2207.03953 · Physical Review A

Nonlinear three-state quantum walks

Pedro R. Nicácio Falcão, J. P. Mendonça, A. R. C. Buarque, Wandearley S. Dias, G. M. A. Almeida, M. L. Lyra

Published Oct 3, 2022• 5 citations
arXivHTMLPDF

Abstract

The dynamics of a three-state quantum walk with amplitude-dependent phase shifts is investigated. We consider two representative inputs whose linear evolution is known to display either full dispersion of the wave packet or intrinsic localization on the initial position. The nonlinear counterpart presents much more involved dynamics featuring self-trapping, solitonic pulses, radiation, and chaoticlike behavior. We show that nonlinearity leads to a metastable self-trapped wave-packet component that radiates in the long-time regime with a survival probability $\ensuremath{\propto}{t}^{\ensuremath{-}1/2}$. A sudden dynamical transition from such a metastable state to the point when the radiation process is triggered is found for a set of parameters.

DOI
10.1103/physreva.106.042202

Related papers

Nonlinear flip-flop quantum walks through potential barriers

F. S. Passos, A. R. C. Buarque

The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent phase shifts with pertubing potential barrier is investigated. Through the adjustment between uniform local perturbations and a Kerr-like nonlinearity of the medium we find a rich set of dynamic profiles. We will show the existence of different Hadamard quantum walking regimes, including those with mobile soliton-like structures or self-trapped states. The latter is predominant for perturbations with amplitudes that tend to $\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath{\pi}/2$. In this system, the qubit shows an unusual behavior as we increase the amplitudes of the potential barriers and displays a monotonic decrease in the self-trapping ${\ensuremath{\varphi}}_{c}$ with respect to the nonlinear parameter. A chaotic-like regime becomes predominant for intermediate nonlinearity values. Furthermore, we show that, by changing the quantum coins ($\ensuremath{\theta}$) a nontrivial dynamic arises, where coins close to Pauli-$X$ drives the system to a regime with predominant soliton-like structures, while the chaotic behavior are restricted to a narrow region in the $\ensuremath{\chi}\ensuremath{-}\ensuremath{\varphi}$ plane. We believe that is possible to implement and observe the proprieties of this model in an integrated photonic system.

arXiv:2210.11661•Physical Review A•Dec 7, 2022•9 citations

Nonlinearity-Inhomogeneity Competition in Discrete-Time Quantum Walks

N. Amaral, A. R. C. Buarque, W. S. Dias

We investigate the interplay between nonlinearity and inhomogeneities in discrete-time quantum walks on one-dimensional lattices. Nonlinear effects are introduced through a Kerr-like, intensity-dependent local phase, while spatial and temporal inhomogeneities are implemented via random variations of the quantum gate operations. By analyzing typical quantities, such as the return probability and the participation function, we identify distinct quantum walking regimes as the nonlinear parameter $χ$ and the quantum gate parameter $θ$ are varied. Spatial inhomogeneities weaken nonlinear self-trapping and constrict the region of robust localization. In this process, partially localized regimes emerge, characterized by the coexistence of a confined core and dispersive wave-packet components. In contrast, temporal inhomogeneities act as time-dependent perturbations that continuously disrupt the phase coherence required for self-trapped excitation, thereby enhancing dispersive emission and promoting delocalization. By using $χ$ versus $θ$ diagrams, we display a comprehensive characterization of how inhomogeneities modify the stability and extent of prevailing dynamical regimes, elucidating the competition between nonlinearity and inhomogeneities in discrete-time quantum walks.

quant-pharXiv:2602.21474•Feb 26, 2026•0 citations

Universal dynamical scaling laws in three-state quantum walks

Pedro R. Nicácio Falcão, A. R. C. Buarque, Wandearley S. Dias +2 more

We perform a finite-time scaling analysis over the detrapping point of a three-state quantum walk on the line. The coin operator is parametrized by ρ that controls the wave packet spreading velocity. The input state prepared at the origin is set as a symmetric linear combination of two eigenstates of the coin operator with a characteristic mixing angle θ, one of them being the component that results in full spreading occurring at θ_{c}(ρ) for which no fraction of the wave packet remains trapped near the initial position. We show that relevant quantities, such as the survival probability and the participation ratio assume single parameter scaling forms at the vicinity of the detrapping angle θ_{c}. In particular, we show that the participation ratio grows linearly in time with a logarithmic correction, thus, shedding light on previous reports of sublinear behavior.

arXiv:2108.10275•Physical review. E•Nov 9, 2021•6 citations

Non-linear Quantum Walks

Carlos Navarrete–Benlloch, Armando Pérez, Eugenio Roldán

We introduce coined Nonlinear Quantum Walks (NLQW) on the lattice. The NLQW is based on the walker acquisition of non-linear (probability dependent) phases at each step of the walk. The most striking result is the formation of non-dispersive pulses in the probability distribution (soliton-like structures). These exhibit a variety of dynamical behaviors, including ballistic motion, dynamical localization, non-elastic collisions and chaotic behavior, in the sense that the dynamics is very sensitive to the nonlinearity strength.

•Apr 12, 2006•0 citations

Emergent nonlinear phenomena in discrete-time quantum walks

J. P. Mendonça, F. A. B. F. de Moura, M. L. Lyra +1 more

Quantum walks are important tools for the development of quantum algorithms and carrying out quantum simulations. Recent interest in nonlinear discrete-time quantum walks aims to use it as a shortcut through dynamical regimes hard to obtain using current methods. We introduce a model featuring a modified conditional shift operator to carry dependence on the local occupation probability with a given strength we are able to control. It accounts for a third-order nonlinear contribution which is found in many physical contexts. We find a rich set of dynamical profiles, including solitonlike propagation, self-trapping, and chaos, all these arising from rather simple rules. Our tool set goes beyond unitary transformations, thus broadening the possibilities for controlling quantum dynamics.

Physical Review A•Jun 29, 2020•19 citations

Self-trapped quantum walks

A. R. C. Buarque, Wandearley S. Dias

We study the existence and characterization of self-trapping phenomena in discrete-time quantum walks. By considering a Kerr-like nonlinearity, we associate an acquisition of the intensity-dependent phase with the walker while it propagates on the lattice. Adjusting the nonlinear parameter $\ensuremath{\chi}$ and the quantum gates $\ensuremath{\theta}$, we will show the existence of different quantum walking regimes, including those with traveling solitonlike structures or localized by self-trapping. The latter scenario is absent for quantum gates close enough to the Pauli-$X$ gate. It appears for intermediate configurations and becomes predominant as quantum gates get closer to the Pauli-$Z$ gate. By using $\ensuremath{\chi}$ versus $\ensuremath{\theta}$ diagrams, we will show that the threshold between quantum walks with delocalized or localized regimes exhibits an unusual aspect in which an increment of the nonlinear strength can induce the system to transition from a localized to a delocalized regime.

arXiv:1911.10831•Physical Review A•Feb 4, 2020•21 citations
The dynamics of a three-state quantum walk with amplitude-dependent phase shifts is investigated. We consider two representative inputs whose linear evolution is known to display either full dispersion of the wave packet or intrinsic localization on the initial position. The nonlinear counterpart presents much more involved dynamics featuring self-trapping, solitonic pulses, radiation, and chaoticlike behavior. We show that nonlinearity leads to a metastable self-trapped wave-packet component that radiates in the long-time regime with a survival probability \ensuremath∝t\ensuremath−1/2\ensuremath{\propto}{t}^{\ensuremath{-}1/2}\ensuremath∝t\ensuremath−1/2. A sudden dynamical transition from such a metastable state to the point when the radiation process is triggered is found for a set of parameters.
The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent phase shifts with pertubing potential barrier is investigated. Through the adjustment between uniform local perturbations and a Kerr-like nonlinearity of the medium we find a rich set of dynamic profiles. We will show the existence of different Hadamard quantum walking regimes, including those with mobile soliton-like structures or self-trapped states. The latter is predominant for perturbations with amplitudes that tend to $\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath{\pi}/2$. In this system, the qubit shows an unusual behavior as we increase the amplitudes of the potential barriers and displays a monotonic decrease in the self-trapping ${\ensuremath{\varphi}}_{c}$ with respect to the nonlinear parameter. A chaotic-like regime becomes predominant for intermediate nonlinearity values. Furthermore, we show that, by changing the quantum coins ($\ensuremath{\theta}$) a nontrivial dynamic arises, where coins close to Pauli-XXX drives the system to a regime with predominant soliton-like structures, while the chaotic behavior are restricted to a narrow region in the $\ensuremath{\chi}\ensuremath{-}\ensuremath{\varphi}$ plane. We believe that is possible to implement and observe the proprieties of this model in an integrated photonic system.
We investigate the interplay between nonlinearity and inhomogeneities in discrete-time quantum walks on one-dimensional lattices. Nonlinear effects are introduced through a Kerr-like, intensity-dependent local phase, while spatial and temporal inhomogeneities are implemented via random variations of the quantum gate operations. By analyzing typical quantities, such as the return probability and the participation function, we identify distinct quantum walking regimes as the nonlinear parameter χχχ and the quantum gate parameter θθθ are varied. Spatial inhomogeneities weaken nonlinear self-trapping and constrict the region of robust localization. In this process, partially localized regimes emerge, characterized by the coexistence of a confined core and dispersive wave-packet components. In contrast, temporal inhomogeneities act as time-dependent perturbations that continuously disrupt the phase coherence required for self-trapped excitation, thereby enhancing dispersive emission and promoting delocalization. By using χχχ versus θθθ diagrams, we display a comprehensive characterization of how inhomogeneities modify the stability and extent of prevailing dynamical regimes, elucidating the competition between nonlinearity and inhomogeneities in discrete-time quantum walks.
We perform a finite-time scaling analysis over the detrapping point of a three-state quantum walk on the line. The coin operator is parametrized by ρ that controls the wave packet spreading velocity. The input state prepared at the origin is set as a symmetric linear combination of two eigenstates of the coin operator with a characteristic mixing angle θ, one of them being the component that results in full spreading occurring at θ_{c}(ρ) for which no fraction of the wave packet remains trapped near the initial position. We show that relevant quantities, such as the survival probability and the participation ratio assume single parameter scaling forms at the vicinity of the detrapping angle θ_{c}. In particular, we show that the participation ratio grows linearly in time with a logarithmic correction, thus, shedding light on previous reports of sublinear behavior.
We introduce coined Nonlinear Quantum Walks (NLQW) on the lattice. The NLQW is based on the walker acquisition of non-linear (probability dependent) phases at each step of the walk. The most striking result is the formation of non-dispersive pulses in the probability distribution (soliton-like structures). These exhibit a variety of dynamical behaviors, including ballistic motion, dynamical localization, non-elastic collisions and chaotic behavior, in the sense that the dynamics is very sensitive to the nonlinearity strength.
Quantum walks are important tools for the development of quantum algorithms and carrying out quantum simulations. Recent interest in nonlinear discrete-time quantum walks aims to use it as a shortcut through dynamical regimes hard to obtain using current methods. We introduce a model featuring a modified conditional shift operator to carry dependence on the local occupation probability with a given strength we are able to control. It accounts for a third-order nonlinear contribution which is found in many physical contexts. We find a rich set of dynamical profiles, including solitonlike propagation, self-trapping, and chaos, all these arising from rather simple rules. Our tool set goes beyond unitary transformations, thus broadening the possibilities for controlling quantum dynamics.
We study the existence and characterization of self-trapping phenomena in discrete-time quantum walks. By considering a Kerr-like nonlinearity, we associate an acquisition of the intensity-dependent phase with the walker while it propagates on the lattice. Adjusting the nonlinear parameter $\ensuremath{\chi}$ and the quantum gates $\ensuremath{\theta}$, we will show the existence of different quantum walking regimes, including those with traveling solitonlike structures or localized by self-trapping. The latter scenario is absent for quantum gates close enough to the Pauli-XXX gate. It appears for intermediate configurations and becomes predominant as quantum gates get closer to the Pauli-ZZZ gate. By using $\ensuremath{\chi}$ versus $\ensuremath{\theta}$ diagrams, we will show that the threshold between quantum walks with delocalized or localized regimes exhibits an unusual aspect in which an increment of the nonlinear strength can induce the system to transition from a localized to a delocalized regime.