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:2512.24616

Arithmetic spectral transition for the unitary almost Mathieu operator

Fan Yang

math.SPmath-phquant-phSubmitted Dec 31, 2025• Announced Jan 1, 2026
arXivHTMLPDF

Abstract

We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov exponent regime $0\le λ_1<λ_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent $β(ω)$. More precisely, for every irrational $ω$ with $β(ω)<L$, where $L>0$ denotes the Lyapunov exponent, and every non-resonant phase $θ$, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which $β(ω)=0$) to a sharp threshold in frequency.

Related papers

Spectral Transitions and Singular Continuous Spectrum in A New Family of Quasi-periodic Quantum Walks

Xinyu Yang, Long Li, Qi Zhou

This paper introduces and rigorously analyzes a new class of one-dimensional discrete-time quantum walks whose dynamics are governed by a parametrized family of extended CMV matrices. The model generalizes the unitary almost Mathieu operator (UAMO) and exhibits a richer spectral phase diagram, closely resembling the extended Harper's model. It provides the first example of a solvable quasi-periodic quantum walk that exhibits a stable region of purely singular continuous spectrum.

quant-pharXiv:2601.20081•Jan 29, 2026

Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

Tatsuya Tate

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.

arXiv:1411.4215•Infinite Dimensional Analysis Quantum Probability and Related Topics•Jun 1, 2019•9 citations

Anderson localization in generalized discrete-time quantum walks

I. Vakulchyk, M. V. Fistul, Pinquan Qin +1 more

We study Anderson localization in a generalized discrete-time quantum walk---a unitary map related to a Floquet driven quantum lattice. It is controlled by a quantum coin matrix which depends on four angles with the meaning of potential and kinetic energy, and external and internal synthetic flux. Such quantum coins can be engineered with microwave pulses in qubit chains. The ordered case yields a two-band eigenvalue structure on the unit circle, which becomes completely flat in the limit of vanishing kinetic energy. Disorder in the external magnetic field does not impact localization. Disorder in all the remaining angles yields Anderson localization. In particular, kinetic-energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. Strong disorder in potential and internal magnetic field energies allows one to obtain analytical expressions for spectrally independent localization length, which is highly useful for various applications.

arXiv:1708.08194•Physical review. B./Physical review. B•Oct 17, 2017•57 citations

Noisy cyclic quantum random walk

G. Juárez Rangel, B. M. Rodríguez-Lara

We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral structure of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bandgaps proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, while non-degenerate spectra produce flat distributions. We introduce static phase noise in the sites and analyze its impact across two distinct propagation regimes. In the walk-on-the-line regime, which precedes a full traversal of the graph, we extract the spreading exponent β from the step-resolved mean squared displacement. We find that low participation ratios correlate with sub-diffusive spread, while high ratios correspond to ballistic or super-diffusive evolution. After the walker completes a cycle, finite-size effects dominate. In this walk-on-the-cycle regime, the spreading exponent no longer characterizes the behavior. We quantify localization using a convergence criterion based on the coefficient of variation of the mean squared displacement. Across both regimes, we identify a sharp crossover near static site noise strength ϕs = π/3. This transition coincides with a drop in the participation ratio, a transition from diffusive to sub-diffusive spread in the walk-on-the-line regime, and a reduction in the saturation level of the mean squared displacement in the walk-on-the-cycle regime. Our results demonstrate that the eigenstate participation ratio provides a computationally efficient spectral diagnostic that anticipates localization across both regimes, offering an alternative to full dynamical simulations.

arXiv:2412.00536•APL Quantum•Aug 8, 2025•0 citations

Observation of Metal-Insulator and Spectral Phase Transitions in Aubry-André-Harper Models

Quan Lin, Christopher Cedzich, Qi Zhou +1 more

Non-Hermitian extensions of the Aubry-André-Harper (AAH) model reveal a rich variety of phase transitions arising from the interplay of quasiperiodicity and nonhermiticity. Despite their theoretical significance, experimental explorations remain challenging due to intricacies in realizing controlled nonhermiticity. Here, we present the first experimental realization of the unitary almost-Mathieu operator that simulates the AAH model by employing single-photon quantum walks. We systematically observe a phase transition between localized and delocalized regimes in the Hermitian limit. Subsequently, by introducing a nonreciprocal hopping, we experimentally probe the parity-time symmetry-breaking transition that is characterized by the emergence of complex quasienergies. Moreover, we identify a novel spectral transition exclusive to discrete-time settings, beyond which the real parts of all quasienergies vanish. These results clarify the interplay between localization, symmetry breaking, and topology in non-Hermitian quasicrystals, thereby paving the way for future exploration of synthetic quantum matter.

arXiv:2508.08255•Physical Review Letters•Apr 30, 2026•1 citation

Anderson localization in a discrete time quantum walk: Interplay of dispersion and disorder

M. V. Fistul, I. Vakulchyk, Pinquan Qin +1 more

A theoretical study of the phenomenon of Anderson localization in a generalized discrete time quantum walk will be reported. The generalized discrete time quantum walk is controlled by a quantum coin unitary matrix on each step of particle motion. The quantum coin matrix depends on four angles that can be engineered with microwave pulses in qubit chains. In the absence of disorder in quantum coins parameters the quantum walk dynamics is determined by two-bands dispersion relationship which becomes completely flat in the limit of vanishing kinetic energy. Various types of disorder result in the localization of quantum particle. In particular, kinetic energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. By making use of the transfer matrix technique we were able to obtain analytical expressions for the frequency dependent localization length in two limits, i.e. weak and strong disorder, and we anticipate that it will be useful for various applications.

AIP conference proceedings•Jan 1, 2018•0 citations
We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on Z2\mathbb Z^2Z2 in a homogeneous magnetic field. In the positive Lyapunov exponent regime 0≤λ1<λ2≤10\le λ_1<λ_2\le 10≤λ1​<λ2​≤1, we establish an arithmetic localization statement governed by the frequency exponent β(ω)β(ω)β(ω). More precisely, for every irrational ωωω with β(ω)<Lβ(ω)<Lβ(ω)<L, where L>0L>0L>0 denotes the Lyapunov exponent, and every non-resonant phase θθθ, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which β(ω)=0β(ω)=0β(ω)=0) to a sharp threshold in frequency.
The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.
We study Anderson localization in a generalized discrete-time quantum walk---a unitary map related to a Floquet driven quantum lattice. It is controlled by a quantum coin matrix which depends on four angles with the meaning of potential and kinetic energy, and external and internal synthetic flux. Such quantum coins can be engineered with microwave pulses in qubit chains. The ordered case yields a two-band eigenvalue structure on the unit circle, which becomes completely flat in the limit of vanishing kinetic energy. Disorder in the external magnetic field does not impact localization. Disorder in all the remaining angles yields Anderson localization. In particular, kinetic-energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. Strong disorder in potential and internal magnetic field energies allows one to obtain analytical expressions for spectrally independent localization length, which is highly useful for various applications.
We explore static noise in a discrete quantum random walk over a homogeneous cyclic graph, focusing on the spectral and dynamical properties of the system. Using a three-parameter unitary coin, we control the spectral structure of the noiseless step operator on the unit circle in the complex plane. One parameter governs the probability amplitudes and induces two spectral bandgaps proportional to its value. The half-sum of the two phase parameters rotates the spectrum and induces twofold degeneracy under specific conditions. Degenerate spectra yield eigenstates with sinusoidal probability distributions, while non-degenerate spectra produce flat distributions. We introduce static phase noise in the sites and analyze its impact across two distinct propagation regimes. In the walk-on-the-line regime, which precedes a full traversal of the graph, we extract the spreading exponent β from the step-resolved mean squared displacement. We find that low participation ratios correlate with sub-diffusive spread, while high ratios correspond to ballistic or super-diffusive evolution. After the walker completes a cycle, finite-size effects dominate. In this walk-on-the-cycle regime, the spreading exponent no longer characterizes the behavior. We quantify localization using a convergence criterion based on the coefficient of variation of the mean squared displacement. Across both regimes, we identify a sharp crossover near static site noise strength ϕs = π/3. This transition coincides with a drop in the participation ratio, a transition from diffusive to sub-diffusive spread in the walk-on-the-line regime, and a reduction in the saturation level of the mean squared displacement in the walk-on-the-cycle regime. Our results demonstrate that the eigenstate participation ratio provides a computationally efficient spectral diagnostic that anticipates localization across both regimes, offering an alternative to full dynamical simulations.
Non-Hermitian extensions of the Aubry-André-Harper (AAH) model reveal a rich variety of phase transitions arising from the interplay of quasiperiodicity and nonhermiticity. Despite their theoretical significance, experimental explorations remain challenging due to intricacies in realizing controlled nonhermiticity. Here, we present the first experimental realization of the unitary almost-Mathieu operator that simulates the AAH model by employing single-photon quantum walks. We systematically observe a phase transition between localized and delocalized regimes in the Hermitian limit. Subsequently, by introducing a nonreciprocal hopping, we experimentally probe the parity-time symmetry-breaking transition that is characterized by the emergence of complex quasienergies. Moreover, we identify a novel spectral transition exclusive to discrete-time settings, beyond which the real parts of all quasienergies vanish. These results clarify the interplay between localization, symmetry breaking, and topology in non-Hermitian quasicrystals, thereby paving the way for future exploration of synthetic quantum matter.
A theoretical study of the phenomenon of Anderson localization in a generalized discrete time quantum walk will be reported. The generalized discrete time quantum walk is controlled by a quantum coin unitary matrix on each step of particle motion. The quantum coin matrix depends on four angles that can be engineered with microwave pulses in qubit chains. In the absence of disorder in quantum coins parameters the quantum walk dynamics is determined by two-bands dispersion relationship which becomes completely flat in the limit of vanishing kinetic energy. Various types of disorder result in the localization of quantum particle. In particular, kinetic energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. By making use of the transfer matrix technique we were able to obtain analytical expressions for the frequency dependent localization length in two limits, i.e. weak and strong disorder, and we anticipate that it will be useful for various applications.