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

International Journal of Mathematical and Computational Methods

Continuous Time Quantum Walks and Recurrences in the Hilbert Space

Alexander Iomin

Published Aug 17, 2016• 1 citation
OpenAlex

Abstract

An analytical consideration of quantum walks in the Hilbert space is suggested for dynamical systems. It is shown that in the semiclassical limit, statistics of the quantum recurrences relates to statistics of the Poincare recurrences of the classical counterpart. It is shown that the statistics of the quantum recurrences is sensitive to the statistics of the corresponding quantum spectrum. The difference in the statistics of quantum recurrences in the Hilbert space for the chaotic and integrable systems follows from the essential difference between the level statistics of integrable and chaotic systems. In particular, when the integrable part of the phase space emerges due to bifurcation, and the exponential distribution of the Poincar´e recurrences of chaotic trajectories is changed into the power law, the statistics of the quantum walks in the Hilbert space follows exactly its classical counterpart.

Related papers

Quantum Walks in Hilbert Space of Lévy Matrices: Recurrences and Revivals

Alexander Iomin

The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown that the statistics of quantum recurrences in the Hilbert space of quantum systems is sensitive to the statistics of the corresponding quantum spectrum. In particular, it is shown that both the Poisson energy level statistics and the Brody distribution correspond to the power law of the quantum recurrences, while the Wigner–Dyson and Lévy–Smirnov statistics of the energy spectra are responsible for the exponential statistics of the quantum returns of the wave function.

Fractal and Fractional•Oct 18, 2021•1 citation

Chaos, Fractal and Quantum Poincare Recurrences in Diamagnetic Kepler Problem

A. Ugulava, L. Chotorlishvili, Tamaz Kereselidze +1 more

The statistics of quantum Poincare recurrences in Hilbert space for diamagnetic hydrogen atom in strong magnetic field has been investigated. It has been shown that quantities characterizing classical chaos are in a good agreement with the ones that are used to describe quantum chaos. The equality of classical and quantum Poincare recurrences has been shown. It has been proved that one of the signs of the emergence of quantum chaos is the irreversible transition from a pure quantum mechanical state to the mixed one.

arXiv:nlin/0608003•Aug 1, 2006•0 citations

Measure theoretical approach to recurrent properties for quantum dynamics

Yoshiki Otobe, Itaru Sasaki

Poincaré's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincaré's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrödinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.

Journal of Physics A Mathematical and Theoretical•Oct 26, 2011•0 citations

Decay Parameter for Continuous-Time Open Quantum Walks

Newton Loebens

Abstract In this paper we study the exponential decay parameter for continuous-time open quantum walks. This statistic is obtained by a composed function that associates the natural logarithm to the transition probability of the walk, being inspired by the classical decay parameter of continuous-time Markov chains. The existence of this statistic is proven for finite graphs where each vertex has a finite degree of freedom, and it is intimately connected to the spectrum of the Lindblad generator describing the dynamics. We show that faithful states yield the slowest decay, whereas non-faithful states can accelerate decay and modify the communication structure of the graph, a phenomenon with no classical analogue. Extensions to infinite graphs are discussed as open problems and conjectures. An equivalence theorem relating different notions of recurrence is established under the hypothesis that the decay parameter exists.

Brazilian Journal of Physics•Jul 8, 2026•0 citations

Chaos and Relaxation in Classical and Quantum Spin Systems

Tarek A. Elsayed

The problems of chaos and relaxation have a fundamental importance in the study of many-body classical and quantum systems. We investigate some of the issues related to these problems numerically in classical and quantum spin systems. New results reported in this thesis include: (i) A remarkably simple algorithm for discriminating chaotic from nonchaotic behavior in classical systems using a time series of one macroscopic observable. The effectiveness of this algorithm stems from the qualitative differences in the power spectra of chaotic and nonchaotic systems. (ii) A modified version of the Onsager regression relation applicable to pure quantum states. (iii) An efficient algorithm for computing the infinitetemperature time correlation functions in systems with large Hilbert spaces. (iv) Absence of exponential sensitivity to small perturbations in macroscopic nonintegrable systems of spins 1/2 . Such a behavior is contrasted with the exponential sensitivity to small perturbations in chaotic classical spin systems. (v) Accurate numerical investigations of free induction decay and spin diffusion in certain spin lattices. The consequences of these results have implications for the foundations of statistical mechanics or practical problems such as computing the long-time behavior of the free induction decay in solids.

•Jan 1, 2013•5 citations

One-dimensional continuous-time quantum walks

Daniel ben‐Avraham, Eric M. Bollt, C. Tamon

We survey the equations of continuous-time quantum walks on simple one-dimensional lattices, which include the finite and infinite lines and the finite cycle, and compare them with the classical continuous-time Markov chains. The focus of our expository article is on analyzing these processes using the Laplace transform on the stochastic recurrences. The resulting time evolution equations, classical versus quantum, are strikingly similar in form, although dissimilar in behavior. We also provide comparisons with analyses performed using spectral methods.

arXiv:cond-mat/0409514•Sep 20, 2004•1 citation
An analytical consideration of quantum walks in the Hilbert space is suggested for dynamical systems. It is shown that in the semiclassical limit, statistics of the quantum recurrences relates to statistics of the Poincare recurrences of the classical counterpart. It is shown that the statistics of the quantum recurrences is sensitive to the statistics of the corresponding quantum spectrum. The difference in the statistics of quantum recurrences in the Hilbert space for the chaotic and integrable systems follows from the essential difference between the level statistics of integrable and chaotic systems. In particular, when the integrable part of the phase space emerges due to bifurcation, and the exponential distribution of the Poincar´e recurrences of chaotic trajectories is changed into the power law, the statistics of the quantum walks in the Hilbert space follows exactly its classical counterpart.
The quantum evolution of wave functions controlled by the spectrum of Lévy random matrices is considered. An analytical treatment of quantum recurrences and revivals in the Hilbert space is performed in the framework of a theory of almost periodic functions. It is shown that the statistics of quantum recurrences in the Hilbert space of quantum systems is sensitive to the statistics of the corresponding quantum spectrum. In particular, it is shown that both the Poisson energy level statistics and the Brody distribution correspond to the power law of the quantum recurrences, while the Wigner–Dyson and Lévy–Smirnov statistics of the energy spectra are responsible for the exponential statistics of the quantum returns of the wave function.
The statistics of quantum Poincare recurrences in Hilbert space for diamagnetic hydrogen atom in strong magnetic field has been investigated. It has been shown that quantities characterizing classical chaos are in a good agreement with the ones that are used to describe quantum chaos. The equality of classical and quantum Poincare recurrences has been shown. It has been proved that one of the signs of the emergence of quantum chaos is the irreversible transition from a pure quantum mechanical state to the mixed one.
Poincaré's recurrence theorem, which states that every Hamiltonian dynamics enclosed in a finite volume returns to its initial position as close as one wishes, is a mathematical basis of statistical mechanics. It is Liouville's theorem that guarantees that the dynamics preserves the volume on the state space. A quantum version of Poincaré's theorem was obtained in the middle of the 20th century without any volume structures of the state space (Hilbert space). One of our aims in this paper is to establish such properties of quantum dynamics from an analog of Liouville's theorem, namely, we will construct a natural probability measure on the Hilbert space from a Hamiltonian defined on the space. Then we will show that the measure is invariant under the corresponding Schrödinger flow. Moreover, we show that the dynamics naturally causes an infinite-dimensional Weyl transformation. It also enables us to discuss the ergodic properties of such dynamics.
Abstract In this paper we study the exponential decay parameter for continuous-time open quantum walks. This statistic is obtained by a composed function that associates the natural logarithm to the transition probability of the walk, being inspired by the classical decay parameter of continuous-time Markov chains. The existence of this statistic is proven for finite graphs where each vertex has a finite degree of freedom, and it is intimately connected to the spectrum of the Lindblad generator describing the dynamics. We show that faithful states yield the slowest decay, whereas non-faithful states can accelerate decay and modify the communication structure of the graph, a phenomenon with no classical analogue. Extensions to infinite graphs are discussed as open problems and conjectures. An equivalence theorem relating different notions of recurrence is established under the hypothesis that the decay parameter exists.
The problems of chaos and relaxation have a fundamental importance in the study of many-body classical and quantum systems. We investigate some of the issues related to these problems numerically in classical and quantum spin systems. New results reported in this thesis include: (i) A remarkably simple algorithm for discriminating chaotic from nonchaotic behavior in classical systems using a time series of one macroscopic observable. The effectiveness of this algorithm stems from the qualitative differences in the power spectra of chaotic and nonchaotic systems. (ii) A modified version of the Onsager regression relation applicable to pure quantum states. (iii) An efficient algorithm for computing the infinitetemperature time correlation functions in systems with large Hilbert spaces. (iv) Absence of exponential sensitivity to small perturbations in macroscopic nonintegrable systems of spins 1/2 . Such a behavior is contrasted with the exponential sensitivity to small perturbations in chaotic classical spin systems. (v) Accurate numerical investigations of free induction decay and spin diffusion in certain spin lattices. The consequences of these results have implications for the foundations of statistical mechanics or practical problems such as computing the long-time behavior of the free induction decay in solids.
We survey the equations of continuous-time quantum walks on simple one-dimensional lattices, which include the finite and infinite lines and the finite cycle, and compare them with the classical continuous-time Markov chains. The focus of our expository article is on analyzing these processes using the Laplace transform on the stochastic recurrences. The resulting time evolution equations, classical versus quantum, are strikingly similar in form, although dissimilar in behavior. We also provide comparisons with analyses performed using spectral methods.