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

Physical Review A

Quantum-fractal revival structure in CN quadratic spectra: Base-Nquantum computer registers

William G. Harter

Published Jun 12, 2001• 20 citations
Publisher

Abstract

A closer examination of Eberly ``revivals,'' Berry ``quantum fractals,'' and Schleich ``carpets'' exposes a simple unitary-cyclic group structure in the dynamics of wave packets for rotorlike systems with quadratic spectra. Quantum quasifractal patterns are revealed to be permeated by self-similar ``X-braid'' structures seen by Hofstadter in Zeeman-perturbed band structures and in other spectra related to circle maps. The patterns are shown to correspond to revival dynamics in the flow of energy or information through these N-level systems with X-braids being directly related to quantum beats or elementary U(2) resonance transfer. ${C}_{N}$-group analysis and Farey arithmetic discretely catalog X-braid hierarchy and suggest the possibility of homocyclic nanostructures or ${C}_{N}$ circuits that are base-N digital quantum computing registers. The phase dynamics corresponds to a fairly sophisticated integer arithmetic that includes an output of all rational numbers resolvable by a given energy input. ${C}_{N}$ circuits automatically give all factors of N. The dynamics of a ${C}_{6}$ device is discussed in detail.

DOI
10.1103/physreva.64.012312

Related papers

Fractional Revival Dynamics in Kerr-Type Systems: Angular Momentum Moments and Classical Analogs

Ashish Kumar Patra, Saikumar Krithivasan

Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.

quant-pharXiv:2601.09763•Jan 15, 2026•0 citations

Unifying Algebraic Structures in Solvable Quantum Models: A Comprehensive Synthesis of Quadratic Tensors, Super-Hopf Algebras, and Fractal-Recursive Ontologies

Syed Muntasir Mamun

This paper presents a rigorous mathematical and ontological synthesis of solvable quantum systems and emergent fractal geometries. By leveraging the quadratic tensor framework, we demonstrate that disparate families of quantum models—including Clifford circuits, stabilizer codes, Gaussian states, and free-fermion systems—can be unified as quadratic functions over abelian groups or super-commutative Hopf algebras. This algebraic unification permits classical simulation with $O(n^2)$ complexity via a generalized Schur complement contraction mechanism, dissolving the boundaries between discrete-variable qudits and continuous-variable bosonic modes. However, physical reality frequently transcends these smooth, integer-dimensional algebraic bases. We integrate a fractal-recursive ontology, identifying that quantum trajectories in path integrals and energy spectra in quantum materials (e.g., Hofstadter's butterfly) exhibit non-integer Hausdorff dimensions and self-similar architectures that challenge traditional linear causality. We further explore the transition from quadratic to $i$-th order tensors, identifying the Clifford hierarchy's correspondence to higher-order polynomial maps and the resulting computational boundaries of the Gottesman-Knill theorem. Also available on Academia.edu: https://www.academia.edu/attachments/131609415/download_file?s=portfolio

•Jan 1, 2026•0 citations

Quantum carpets: A probe to identify wave packet fractional revivals

Iqra Yausaf, Shahid Iqbal

Quantum carpets- in position and momentum space- woven by the self-interference of de Broglie wave of an atom or an electron, trapped in an infinitely deep potential well, are explained. The recurrence of self-similar structures in designs of these carpets mimics the phenomena of quantum revivals and fractional revivals. We identify fractional revivals of various order by means of these space-time and momentum-time interference patterns.

arXiv:1607.07496•Jul 25, 2016•0 citations

Quantum carpets: efficiently probing fractional revivals in position-dependent mass systems

Tooba Bibi, Sunia Javed, Shahid Iqbal

Abstract Position-dependent-mass systems are of great importance in many physical situations, such as the transport of charge carriers in semiconductors with non-uniform composition and in the theory of many-body interactions in condensed matter. Here we investigate, numerically and analytically, the phenomenon of fractional revivals in such systems, which is a generic characteristic manifested by the wave-packet evolution in bounded Hamiltonian systems. Identifying the fractional revivals using specific probes is an important task in the theory of quantum measurement and sensing. We numerically simulate the temporal evolution of probability density and information entropy density, which manifest self-similarly recurring interference patterns, namely, quantum carpets. Our numerical results show that the quantum carpets not only serve as an effective probe for recognizing the fractional revivals of various order but they efficiently describe the effect of spatially-varying mass on the structure of fractional revivals, which is manifested as a symmetry breaking in their designs.

Communications in Theoretical Physics•Nov 1, 2022•3 citations

Wave packet dynamics and factorization of numbers

Holger Mack, Marc Bienert, Florian Haug +3 more

We connect three phenomena of wave packet dynamics: Talbot images, revivals of a particle in a box and fractional revivals. The physical origin of these effects is deeply rooted in phase factors which are quadratic in the quantum number. We show that the characteristic structures in the time evolution of these systems allow us to factorize large integers.

arXiv:quant-ph/0204040•Apr 8, 2002•2 citations

Quantum revivals, geometric phases and circle map recurrences

Seshadri, S R, Lakshmibala, S, Balakrishnan, V

Revivals of the coherent states of a deformed, adiabatically and cyclically varying oscillator Hamiltonian are examined. The revival time distribution is exactly that of Poincaré recurrences for a rotation map: only three distinct revival times can occur, with specified weights. A link is thus established between quantum revivals and recurrences in a coarse-grained discrete-time dynamical system.

•Jan 1, 1999•6 citations
A closer examination of Eberly revivals,'' Berry quantum fractals,'' and Schleich carpets'' exposes a simple unitary-cyclic group structure in the dynamics of wave packets for rotorlike systems with quadratic spectra. Quantum quasifractal patterns are revealed to be permeated by self-similar X-braid'' structures seen by Hofstadter in Zeeman-perturbed band structures and in other spectra related to circle maps. The patterns are shown to correspond to revival dynamics in the flow of energy or information through these N-level systems with X-braids being directly related to quantum beats or elementary U(2) resonance transfer. CN{C}_{N}CN​-group analysis and Farey arithmetic discretely catalog X-braid hierarchy and suggest the possibility of homocyclic nanostructures or CN{C}_{N}CN​ circuits that are base-N digital quantum computing registers. The phase dynamics corresponds to a fairly sophisticated integer arithmetic that includes an output of all rational numbers resolvable by a given energy input. CN{C}_{N}CN​ circuits automatically give all factors of N. The dynamics of a C6{C}_{6}C6​ device is discussed in detail.
Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.
This paper presents a rigorous mathematical and ontological synthesis of solvable quantum systems and emergent fractal geometries. By leveraging the quadratic tensor framework, we demonstrate that disparate families of quantum models—including Clifford circuits, stabilizer codes, Gaussian states, and free-fermion systems—can be unified as quadratic functions over abelian groups or super-commutative Hopf algebras. This algebraic unification permits classical simulation with O(n2)O(n^2)O(n2) complexity via a generalized Schur complement contraction mechanism, dissolving the boundaries between discrete-variable qudits and continuous-variable bosonic modes. However, physical reality frequently transcends these smooth, integer-dimensional algebraic bases. We integrate a fractal-recursive ontology, identifying that quantum trajectories in path integrals and energy spectra in quantum materials (e.g., Hofstadter's butterfly) exhibit non-integer Hausdorff dimensions and self-similar architectures that challenge traditional linear causality. We further explore the transition from quadratic to iii-th order tensors, identifying the Clifford hierarchy's correspondence to higher-order polynomial maps and the resulting computational boundaries of the Gottesman-Knill theorem. Also available on Academia.edu: https://www.academia.edu/attachments/131609415/download_file?s=portfolio
Abstract Position-dependent-mass systems are of great importance in many physical situations, such as the transport of charge carriers in semiconductors with non-uniform composition and in the theory of many-body interactions in condensed matter. Here we investigate, numerically and analytically, the phenomenon of fractional revivals in such systems, which is a generic characteristic manifested by the wave-packet evolution in bounded Hamiltonian systems. Identifying the fractional revivals using specific probes is an important task in the theory of quantum measurement and sensing. We numerically simulate the temporal evolution of probability density and information entropy density, which manifest self-similarly recurring interference patterns, namely, quantum carpets. Our numerical results show that the quantum carpets not only serve as an effective probe for recognizing the fractional revivals of various order but they efficiently describe the effect of spatially-varying mass on the structure of fractional revivals, which is manifested as a symmetry breaking in their designs.