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:2610.00712

Krylov complexity of a tilted extended Bose-Hubbard chain with Rydberg-dressed interactions

Yifan Chen, Tianyi Yan, Lu Qin, Weibin Li

Published Sep 30, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate Krylov state complexity in a tilted extended Bose-Hubbard chain in which both the on-site interaction $U$ and nearest-neighbor interaction $V$ are present. The tilted extended Bose-Hubbard model can be realized with Rydberg-dressed interactions in optical lattices. Using exact diagonalization and Lanczos recursion from three physically motivated Fock states, we compute the time-dependent complexity, its long-time saturation value, and the fluctuations of the Lanczos coefficients for an open chain at unit filling. Crucially, we identify a pronounced quasi-chaotic diagonal regime near $U\simeq V$ in which the Krylov complexity saturation exhibits a sharp ridge while the spectral level-spacing statistics remain partially chaotic. This dissociation arises from the competition between the on-site and nearest-neighbor interaction, which generates a dense manifold of near-resonant many-body configurations that is efficiently coupled by the state-dependent dynamics but retains residual spectral structure. Our results show that Krylov state complexity provides complementary, state-sensitive information about many-body chaos that is not captured by the spectral diagnostics alone, establishing Krylov complexity as a useful diagnostic for studying thermalization and information scrambling in experimentally accessible quantum simulators.

DOI
10.48550/arxiv.2610.00712

Related papers

Operator growth and Krylov complexity in Bose-Hubbard model

Arpan Bhattacharyya, Debodirna Ghosh, Poulami Nandi

Abstract We study Krylov complexity of a one-dimensional Bosonic system, the celebrated Bose-Hubbard Model. The Bose-Hubbard Hamiltonian consists of interacting bosons on a lattice, describing ultra-cold atoms. Apart from showing superfluid-Mott insulator phase transition, the model also exhibits both chaotic and integrable (mixed) dynamics depending on the value of the interaction parameter. We focus on the three-site Bose Hubbard Model (with different particle numbers), which is known to be highly mixed. We use the Lanczos algorithm to find the Lanczos coefficients and the Krylov basis. The orthonormal Krylov basis captures the operator growth for a system with a given Hamiltonian. However, the Lanczos algorithm needs to be modified for our case due to the instabilities instilled by the piling up of computational errors. Next, we compute the Krylov complexity and its early and late-time behaviour. Our results capture the chaotic and integrable nature of the system. Our paper takes the first step to use the Lanczos algorithm non-perturbatively for a discrete quartic bosonic Hamiltonian without depending on the auto-correlation method.

arXiv:2306.05542•Journal of High Energy Physics•Dec 18, 2023•30 citations

Unfolded Krylov complexity: universal chaotic dynamics without false positives

Johanna Erdmenger, Kyoung-Bum Huh, Hyun-Sik Jeong +1 more

A central challenge in diagnosing quantum chaos is to distinguish genuine many-body scrambling from kinematic effects of the spectrum. Krylov state complexity, or spread complexity, has emerged as a powerful diagnostic, with its characteristic growth, peak, and relaxation often taken as signatures of chaos. However, previous work has shown that this criterion can give false positives: saddle-dominated integrable systems may display prominent peaks even without random-matrix level correlations. We argue, based on complementary numerical and analytical evidence, that this ambiguity can be resolved by unfolding the spectrum prior to constructing the ensuing Krylov dynamics. By removing the non-universal smooth density of states while retaining microscopic spectral correlations, unfolding suppresses spurious peaks in integrable systems while preserving the universal spectral signatures of chaotic systems. Analytically, the formulation of the Lanczos iteration in terms of orthogonal polynomials yields an exact complexity kernel with a robust near-diagonal structure whose fine-grained features reflect the underlying spectral correlations. Moreover, for the logarithmic model, unfolding can be performed exactly, mapping the spectrum to a uniform lattice and yielding an analytic spread complexity that removes the false-positive peak. These findings establish unfolded Krylov complexity as a more reliable probe of genuine many-body scrambling.

hep-thCross-listedarXiv:2609.02228•Physical review. D/Physical review. D.•Sep 3, 2026•0 citations

Quantum Chaos and Spread of States in Krylov Subspace: A Topical Review

Hyun-Sik Jeong, Juan F. Pedraza

Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.

hep-thCross-listedarXiv:2608.26825•Aug 28, 2026•0 citations

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Aranya Bhattacharya, Pratik Nandy, Pingal Pratyush Nath +1 more

A bstract Continuing the previous initiatives [1, 2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and non-integrable systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

arXiv:2303.04175•Journal of High Energy Physics•Dec 12, 2023•82 citations

Realization of a Rydberg-dressed extended Bose-Hubbard model

Pascal Weckesser, Kritsana Srakaew, Tizian Blatz +6 more

The competition of different length scales in quantum many-body systems leads to phenomena such as correlated dynamics and nonlocal order. To investigate such effects in an itinerant lattice-based quantum simulator, it has been proposed to introduce tunable extended-range interactions using off-resonant optical coupling to Rydberg states, known as Rydberg dressing. In this work, we use this approach to realize an effective one-dimensional extended Bose-Hubbard model. Harnessing our quantum gas microscope, we probe the correlated out-of-equilibrium dynamics of extended-range repulsively bound pairs and "hard rods." By contrast, operating near equilibrium, we observe density ordering when adiabatically turning on the extended-range interactions. Our results pave the way to realizing light-controlled extended-range interacting quantum many-body systems.

arXiv:2405.20128•Science•Nov 20, 2025•14 citations

Krylov complexity of anyons

Ya-Nan Wang, Qing-Min Hu, Wen-Long You +1 more

Anyons obey fractional statistics that lie between bosonic and fermionic statistics, giving rise to a broad range of intriguing phenomena. However, how anyonic statistics govern quantum-state complexity is still largely unexplored. In this work, we investigate the interplay between the statistical phase and on-site interactions in the anyon-Hubbard model, identifying exact quantum many-body scar eigenstates and novel quench dynamics. The Krylov complexity exhibits perfect periodic revivals independent of the statistical phase in the scarred dynamics, whereas after a quench it depends on both the statistical phase and the interaction strength. In the strong-interaction regime, we find approximate scarred dynamics, while in the weak-interaction regime the state spreads over Krylov space and the complexity ultimately saturates. Moreover, for the bosonic initial state, the complexity of fermions exhibits the lowest saturation value, and vice versa. For fractional statistics, the saturation plateau is minimized when the post-quench statistical phase is close to that of the initial state. Our results demonstrate the central role of the statistical phase in governing many-body dynamics and provide new insights into Krylov complexity and quantum many-body scars.

quant-pharXiv:2608.16595•Aug 18, 2026•0 citations
We investigate Krylov state complexity in a tilted extended Bose-Hubbard chain in which both the on-site interaction UUU and nearest-neighbor interaction VVV are present. The tilted extended Bose-Hubbard model can be realized with Rydberg-dressed interactions in optical lattices. Using exact diagonalization and Lanczos recursion from three physically motivated Fock states, we compute the time-dependent complexity, its long-time saturation value, and the fluctuations of the Lanczos coefficients for an open chain at unit filling. Crucially, we identify a pronounced quasi-chaotic diagonal regime near U≃VU\simeq VU≃V in which the Krylov complexity saturation exhibits a sharp ridge while the spectral level-spacing statistics remain partially chaotic. This dissociation arises from the competition between the on-site and nearest-neighbor interaction, which generates a dense manifold of near-resonant many-body configurations that is efficiently coupled by the state-dependent dynamics but retains residual spectral structure. Our results show that Krylov state complexity provides complementary, state-sensitive information about many-body chaos that is not captured by the spectral diagnostics alone, establishing Krylov complexity as a useful diagnostic for studying thermalization and information scrambling in experimentally accessible quantum simulators.
Abstract We study Krylov complexity of a one-dimensional Bosonic system, the celebrated Bose-Hubbard Model. The Bose-Hubbard Hamiltonian consists of interacting bosons on a lattice, describing ultra-cold atoms. Apart from showing superfluid-Mott insulator phase transition, the model also exhibits both chaotic and integrable (mixed) dynamics depending on the value of the interaction parameter. We focus on the three-site Bose Hubbard Model (with different particle numbers), which is known to be highly mixed. We use the Lanczos algorithm to find the Lanczos coefficients and the Krylov basis. The orthonormal Krylov basis captures the operator growth for a system with a given Hamiltonian. However, the Lanczos algorithm needs to be modified for our case due to the instabilities instilled by the piling up of computational errors. Next, we compute the Krylov complexity and its early and late-time behaviour. Our results capture the chaotic and integrable nature of the system. Our paper takes the first step to use the Lanczos algorithm non-perturbatively for a discrete quartic bosonic Hamiltonian without depending on the auto-correlation method.
A central challenge in diagnosing quantum chaos is to distinguish genuine many-body scrambling from kinematic effects of the spectrum. Krylov state complexity, or spread complexity, has emerged as a powerful diagnostic, with its characteristic growth, peak, and relaxation often taken as signatures of chaos. However, previous work has shown that this criterion can give false positives: saddle-dominated integrable systems may display prominent peaks even without random-matrix level correlations. We argue, based on complementary numerical and analytical evidence, that this ambiguity can be resolved by unfolding the spectrum prior to constructing the ensuing Krylov dynamics. By removing the non-universal smooth density of states while retaining microscopic spectral correlations, unfolding suppresses spurious peaks in integrable systems while preserving the universal spectral signatures of chaotic systems. Analytically, the formulation of the Lanczos iteration in terms of orthogonal polynomials yields an exact complexity kernel with a robust near-diagonal structure whose fine-grained features reflect the underlying spectral correlations. Moreover, for the logarithmic model, unfolding can be performed exactly, mapping the spectrum to a uniform lattice and yielding an analytic spread complexity that removes the false-positive peak. These findings establish unfolded Krylov complexity as a more reliable probe of genuine many-body scrambling.
Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.
A bstract Continuing the previous initiatives [1, 2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and non-integrable systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.
The competition of different length scales in quantum many-body systems leads to phenomena such as correlated dynamics and nonlocal order. To investigate such effects in an itinerant lattice-based quantum simulator, it has been proposed to introduce tunable extended-range interactions using off-resonant optical coupling to Rydberg states, known as Rydberg dressing. In this work, we use this approach to realize an effective one-dimensional extended Bose-Hubbard model. Harnessing our quantum gas microscope, we probe the correlated out-of-equilibrium dynamics of extended-range repulsively bound pairs and "hard rods." By contrast, operating near equilibrium, we observe density ordering when adiabatically turning on the extended-range interactions. Our results pave the way to realizing light-controlled extended-range interacting quantum many-body systems.
Anyons obey fractional statistics that lie between bosonic and fermionic statistics, giving rise to a broad range of intriguing phenomena. However, how anyonic statistics govern quantum-state complexity is still largely unexplored. In this work, we investigate the interplay between the statistical phase and on-site interactions in the anyon-Hubbard model, identifying exact quantum many-body scar eigenstates and novel quench dynamics. The Krylov complexity exhibits perfect periodic revivals independent of the statistical phase in the scarred dynamics, whereas after a quench it depends on both the statistical phase and the interaction strength. In the strong-interaction regime, we find approximate scarred dynamics, while in the weak-interaction regime the state spreads over Krylov space and the complexity ultimately saturates. Moreover, for the bosonic initial state, the complexity of fermions exhibits the lowest saturation value, and vice versa. For fractional statistics, the saturation plateau is minimized when the post-quench statistical phase is close to that of the initial state. Our results demonstrate the central role of the statistical phase in governing many-body dynamics and provide new insights into Krylov complexity and quantum many-body scars.