© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2604.27851

Wavelet-based multiresolution analysis of quantum fractals in confined dynamics

David Navia, Ángel S. Sanz

quant-phSubmitted Apr 30, 2026• Announced May 1, 2026• 0 citations

Abstract

Fractal structures naturally emerge in quantum systems whose initial states exhibit spatial discontinuities, a phenomenon first identified by Berry in the paradigmatic case of a particle confined in an infinite potential well. While previous analyses of quantum fractals have mainly relied on spectral decompositions and geometric scaling arguments, their quantitative characterization often depends on scale choices and truncation effects. Here we present a wavelet-based multiresolution framework that enables a direct and assumption-free quantification of quantum fractality. Fractal dimensions are extracted from the scale-dependent distribution of wavelet energies, without invoking prior power-law hypotheses. The method is applied to space and time quantum fractals arising in confined dynamics, as well as to dynamical curves generated by the associated quantum probability flux. These flux-driven trajectories provide a natural space--time parametrization of the underlying fractal structure and yield scaling properties fully consistent with Berry's predictions for space--time fractals. The resulting fractal dimensions are shown to be robust with respect to the choice of wavelet family, numerical cutoffs, and system parameters. Beyond validating earlier conjectures, the present framework offers a unified and computationally efficient tool for the multiscale analysis of quantum fractality in confined and interference-driven quantum dynamics. That is, it provides an operational, scale-adaptive criterion that unifies the characterization of space, time, and space--time quantum fractals within a single, hypothesis-free approach.

Comments: 6 pages, 4 figures, Supplemental Material

Related papers

Quantum Mechanical Operators in Multiresolution Hilbert Spaces

Jānos Pipek, Theodore E. Simos, George Maroulis

Wavelet analysis, which is a shorthand notation for the concept of multiresolution analysis (MRA), becomes increasingly popular in high efficiency storage algorithms of complex spatial distributions. This approach is applied for describing wave functions of quantum systems. At any resolution level of MRA expansions a physical observable is represented by an infinite matrix which is “canonically” chosen as the projection of its operator in the Schrödinger picture onto the subspace of the given resolution. It is shown that this canonical choice is only a particular member of possible operator representations. Among these, there exits an optimal choice, usually different from the canonical one, which gives the best numerical values in eigenvalue problems. This construction works even in those cases, where the canonical definition is unusable. The commutation relation of physical operators is also studied in MRA subspaces. It is shown that the required commutation rules are satisfied in the fine resolution limit, whereas in coarse grained spaces a correction appears depending only on the representation of the momentum operator.

AIP conference proceedings•Jan 1, 2007•0 citations

Quantum Nondecimated Wavelet Transform: Theory, Circuits, and Applications

Brani Vidakovic

The nondecimated or translation-invariant wavelet transform (NDWT) is a central tool in classical multiscale signal analysis, valued for its stability, redundancy, and shift invariance. This paper develops two complementary quantum formulations of the NDWT that embed these classical properties coherently into quantum computation. The first formulation is based on the epsilon-decimated interpretation of the NDWT and realizes all circularly shifted wavelet transforms simultaneously by promoting the shift index to a quantum register and applying controlled circular shifts followed by a wavelet analysis unitary. The resulting construction yields an explicit, fully unitary quantum representation of redundant wavelet coefficients and supports coherent postprocessing, including quantum shrinkage via ancilla-driven completely positive trace preserving maps. The second formulation is based on the Hadamard test and uses diagonal phase operators to probe scale-shift wavelet structure through interference, providing direct access to shift-invariant energy scalograms and multiscale spectra without explicit coefficient reconstruction. Together, these two approaches demonstrate that redundancy and translation invariance can be exploited rather than avoided in the quantum setting. Applications to denoising, feature extraction, and spectral scaling illustrate how quantum NDWTs provide a flexible and physically meaningful foundation for multiscale quantum signal processing.

quant-pharXiv:2512.21478•Dec 26, 2025•0 citations

Quantum fractals in boxes

Michael Berry

A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .

Journal of Physics A Mathematical and General•Oct 21, 1996•193 citations

Multiresolution decomposition of quantum field theories using wavelet bases

Tracie Michlin, Wayne Nicholas Polyzou, Fatih Bulut

We investigate both the theoretical and computational aspects of using wavelet bases to perform an exact decomposition of a local field theory by spatial resolution. The decomposition admits natural volume and resolution truncations. We demonstrate that flow equation methods can be used to eliminate short-distance degrees of freedom in truncated theories. The method is tested on a free scalar field in one dimension, where the spatial derivatives couple the degrees of freedom on different scales, although the method is applicable to more complex field theories. The flow equation method is shown to decouple both distance and energy scales in this example. The response to changing the volume and resolution cutoffs and the mass is discussed.

arXiv:1610.08612•Physical review. D/Physical review. D.•May 1, 2017•11 citations

Quantum many-body operator cascade as a route to chaos

Urban Duh, Marko Žnidarič

We describe a chaos scenario valid for chaotic many-body quantum systems, the simulation of which is one of the prime candidates for demonstrating quantum advantage, and clarify its importance for classical truncation-based simulations. By studying the spectral properties of the truncated propagator, we, for the first time, identify fractal structures responsible for many-body quantum chaos, analogous to the fractality in classical chaos emerging from the time evolution of initially smooth long-wavelength densities into ever finer short-wavelength densities. Namely, we show that the leading slowest-decaying Ruelle-Pollicott eigenoperators have a non-trivial fractal dimension quantifying their non-locality, visible also in the divergence of their condition numbers. Furthermore, we find that unitarity imposes an asymptotic equality between the temporal decay rate of local correlations and this many-body fractal dimension. With this insight, the scenario for many-body quantum chaos becomes clear: over time, local operators evolve toward increasingly non-local ones with a quantifiable fractal structure, thereby naturally leading to effective non-unitary relaxation on the subspace of local operators -- a kind of many-body Kolmogorov cascade in the space of operators -- which enables accurate truncation-based simulations. Our predictions are demonstrated numerically in various one- and two-dimensional quantum circuits, with and without conservation laws, as well as by exact results in dual-unitary circuits.

cond-mat.stat-mechCross-listedarXiv:2604.16720•Apr 21, 2026•0 citations

Multiscale quantum propagation using compact-support wavelets in space and time

Haixiang Wang, Ramiro Acevedo, Heather Mollé +3 more

Orthogonal compact-support Daubechies wavelets are employed as bases for both space and time variables in the solution of the time-dependent Schrodinger equation. Initial value conditions are enforced using special early-time wavelets analogous to edge wavelets used in boundary-value problems. It is shown that the quantum equations may be solved directly and accurately in the discrete wavelet representation, an important finding for the eventual goal of highly adaptive multiresolution Schrodinger equation solvers. While the temporal part of the basis is not sharp in either time or frequency, the Chebyshev method used for pure time-domain propagations is adapted to use in the mixed domain and is able to take advantage of Hamiltonian matrix sparseness. The orthogonal separation into different time scales is determined theoretically to persist throughout the evolution and is demonstrated numerically in a partially adaptive treatment of scattering from an asymmetric Eckart barrier.

The Journal of Chemical Physics•Oct 12, 2004•7 citations

Fractal structures naturally emerge in quantum systems whose initial states exhibit spatial discontinuities, a phenomenon first identified by Berry in the paradigmatic case of a particle confined in an infinite potential well. While previous analyses of quantum fractals have mainly relied on spectral decompositions and geometric scaling arguments, their quantitative characterization often depends on scale choices and truncation effects. Here we present a wavelet-based multiresolution framework that enables a direct and assumption-free quantification of quantum fractality. Fractal dimensions are extracted from the scale-dependent distribution of wavelet energies, without invoking prior power-law hypotheses. The method is applied to space and time quantum fractals arising in confined dynamics, as well as to dynamical curves generated by the associated quantum probability flux. These flux-driven trajectories provide a natural space--time parametrization of the underlying fractal structure and yield scaling properties fully consistent with Berry's predictions for space--time fractals. The resulting fractal dimensions are shown to be robust with respect to the choice of wavelet family, numerical cutoffs, and system parameters. Beyond validating earlier conjectures, the present framework offers a unified and computationally efficient tool for the multiscale analysis of quantum fractality in confined and interference-driven quantum dynamics. That is, it provides an operational, scale-adaptive criterion that unifies the characterization of space, time, and space--time quantum fractals within a single, hypothesis-free approach.

Wavelet analysis, which is a shorthand notation for the concept of multiresolution analysis (MRA), becomes increasingly popular in high efficiency storage algorithms of complex spatial distributions. This approach is applied for describing wave functions of quantum systems. At any resolution level of MRA expansions a physical observable is represented by an infinite matrix which is “canonically” chosen as the projection of its operator in the Schrödinger picture onto the subspace of the given resolution. It is shown that this canonical choice is only a particular member of possible operator representations. Among these, there exits an optimal choice, usually different from the canonical one, which gives the best numerical values in eigenvalue problems. This construction works even in those cases, where the canonical definition is unusable. The commutation relation of physical operators is also studied in MRA subspaces. It is shown that the required commutation rules are satisfied in the fine resolution limit, whereas in coarse grained spaces a correction appears depending only on the representation of the momentum operator.

The nondecimated or translation-invariant wavelet transform (NDWT) is a central tool in classical multiscale signal analysis, valued for its stability, redundancy, and shift invariance. This paper develops two complementary quantum formulations of the NDWT that embed these classical properties coherently into quantum computation. The first formulation is based on the epsilon-decimated interpretation of the NDWT and realizes all circularly shifted wavelet transforms simultaneously by promoting the shift index to a quantum register and applying controlled circular shifts followed by a wavelet analysis unitary. The resulting construction yields an explicit, fully unitary quantum representation of redundant wavelet coefficients and supports coherent postprocessing, including quantum shrinkage via ancilla-driven completely positive trace preserving maps. The second formulation is based on the Hadamard test and uses diagonal phase operators to probe scale-shift wavelet structure through interference, providing direct access to shift-invariant energy scalograms and multiscale spectra without explicit coefficient reconstruction. Together, these two approaches demonstrate that redundancy and translation invariance can be exploited rather than avoided in the quantum setting. Applications to denoising, feature extraction, and spectral scaling illustrate how quantum NDWTs provide a flexible and physically meaningful foundation for multiscale quantum signal processing.

A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .

We investigate both the theoretical and computational aspects of using wavelet bases to perform an exact decomposition of a local field theory by spatial resolution. The decomposition admits natural volume and resolution truncations. We demonstrate that flow equation methods can be used to eliminate short-distance degrees of freedom in truncated theories. The method is tested on a free scalar field in one dimension, where the spatial derivatives couple the degrees of freedom on different scales, although the method is applicable to more complex field theories. The flow equation method is shown to decouple both distance and energy scales in this example. The response to changing the volume and resolution cutoffs and the mass is discussed.

We describe a chaos scenario valid for chaotic many-body quantum systems, the simulation of which is one of the prime candidates for demonstrating quantum advantage, and clarify its importance for classical truncation-based simulations. By studying the spectral properties of the truncated propagator, we, for the first time, identify fractal structures responsible for many-body quantum chaos, analogous to the fractality in classical chaos emerging from the time evolution of initially smooth long-wavelength densities into ever finer short-wavelength densities. Namely, we show that the leading slowest-decaying Ruelle-Pollicott eigenoperators have a non-trivial fractal dimension quantifying their non-locality, visible also in the divergence of their condition numbers. Furthermore, we find that unitarity imposes an asymptotic equality between the temporal decay rate of local correlations and this many-body fractal dimension. With this insight, the scenario for many-body quantum chaos becomes clear: over time, local operators evolve toward increasingly non-local ones with a quantifiable fractal structure, thereby naturally leading to effective non-unitary relaxation on the subspace of local operators -- a kind of many-body Kolmogorov cascade in the space of operators -- which enables accurate truncation-based simulations. Our predictions are demonstrated numerically in various one- and two-dimensional quantum circuits, with and without conservation laws, as well as by exact results in dual-unitary circuits.

Orthogonal compact-support Daubechies wavelets are employed as bases for both space and time variables in the solution of the time-dependent Schrodinger equation. Initial value conditions are enforced using special early-time wavelets analogous to edge wavelets used in boundary-value problems. It is shown that the quantum equations may be solved directly and accurately in the discrete wavelet representation, an important finding for the eventual goal of highly adaptive multiresolution Schrodinger equation solvers. While the temporal part of the basis is not sharp in either time or frequency, the Chebyshev method used for pure time-domain propagations is adapted to use in the mixed domain and is able to take advantage of Hamiltonian matrix sparseness. The orthogonal separation into different time scales is determined theoretically to persist throughout the evolution and is demonstrated numerically in a partially adaptive treatment of scattering from an asymmetric Eckart barrier.