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:2607.10454

Krylov Complexity for Time-Dependent Hamiltonians

Amin Faraji Astaneh, Parsa Kafashi

hep-thcond-mat.stat-mechmath-phquant-phSubmitted Jul 11, 2026• Announced Jul 14, 2026• 0 citations• Revised Jul 15, 2026 (v2)
v1 · Jul 11, 2026v2 · Jul 15, 2026
arXivHTMLPDF

Abstract

We investigate Krylov spread complexity for states evolving under time-dependent Hamiltonians. For periodically driven systems, we formulate the problem within Floquet theory and show how the Magnus expansion provides a systematic approximation when the Floquet Hamiltonian is not available in closed form. We then extend this framework beyond periodic driving and demonstrate that, in addition to the globally truncated Magnus expansion, a piecewise Magnus expansion provides a reliable method when the global expansion loses convergence or accuracy. Our results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems.

AI Summary

The paper studies how to compute state Krylov (spread) complexity under explicitly time-dependent Hamiltonians using effective generators obtained from Floquet theory and Magnus expansions. It treats periodically driven two-level systems and a harmonic oscillator with time-dependent frequency. For periodic driving, the Krylov basis is constructed from a Floquet Hamiltonian, giving a stroboscopic complexity. For $H_F=\mathbf h\cdot\boldsymbol\sigma$ and an initial $\sigma_z$ eigenstate, $C_K(t)=(h_x^2+h_y^2)\sin^2(|\mathbf h|t)/|\mathbf h|^2$. Circular driving is solved exactly in a rotating frame; linear driving is treated near resonance by the rotating-wave approximation and at high frequency by a dressed-frame Magnus expansion. The latter produces a Bessel-renormalized longitudinal field and a second-order transverse correction involving Bessel and Struve functions. For the oscillator, symplectic propagation determines Bogoliubov coefficients. In the chosen even-Fock Krylov chain, $C_K(t)=|\nu(t)|^2/2$. A sudden quench gives an exact oscillatory expression; smooth quenches are computed numerically by composing local Magnus propagators, recovering sudden-quench and approximately adiabatic behavior. The paper reviews the sufficient convergence condition $\int_0^t\|H(s)\|ds<\pi$. Piecewise truncation at order $m$ has stated accumulated error $O(t\Delta^m)$, assuming controlled stability factors. Limitations include global Magnus convergence and long-time accuracy, accumulated local errors, quasienergy-branch choices, and micromotion outside stroboscopic times. Near zeros of the leading Bessel term, higher Magnus orders determine residual dynamics. Extensions beyond the examples and to operator complexity remain future directions.

Likely largely AI-generatedIncrementalUnclear noveltyMissing comparisonStrong assumptions

Error found: The soft-quench examples do not directly compare global and piecewise Magnus approximations in a regime where the global approximation fails. The reported soft-quench calculations omit step sizes, quantitative numerical errors, and refinement tests, leaving their claimed numerical reliability insufficiently documented. The oscillator step-size prescription reuses a Hamiltonian operator norm that is infinite for the harmonic oscillator, rather than explicitly establishing convergence using the finite-dimensional phase-space generator.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Comments
27 pages, 4 figures, Refs. updated

Related papers

A Variational Framework for Time-Dependent Quantum Systems with Applications to Floquet Hamiltonians

Ibsal Assi, Meenu Kumari, J. P. F. LeBlanc

We introduce a variational framework for approximating the time-evolution operator $\hat{U}(t)$ within a physically motivated operator manifold, reformulating quantum dynamics as a tractable problem in operator space using stationary action principle. For periodically driven systems, the resulting approximate evolution operator directly yields an effective Floquet Hamiltonian, offering a non-perturbative alternative to conventional expansion-based methods. The framework is systematically improvable by enlarging the operator pool and naturally incorporates symmetries and physical constraints. When the operator manifold is chosen from the terms of a truncated Magnus expansion, the variational procedure effectively resums the Magnus series within the restricted space, significantly enhancing accuracy. We benchmark the approach on the driven Rabi model, the driven Lipkin-Meshkov-Glick model, and the one-dimensional driven Ising chain, yielding effective Floquet Hamiltonians that are systematically more accurate than low-order Magnus expansions, particularly in regimes where the latter converge poorly, and illustrating applicability to systems with exponentially large Hilbert spaces. Although we focus here on Floquet systems, the formalism applies equally to generic time-dependent Hamiltonians, providing a versatile tool for non-equilibrium quantum dynamics.

quant-pharXiv:2607.29418•Aug 3, 2026•0 citations

The Floquet-Magnus expansion of unbounded operators

Daniel Burgarth, Robin Hillier, Davide Lonigro +1 more

The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.

math-phCross-listedarXiv:2605.23734•May 25, 2026

Analytic approaches to periodically driven closed quantum systems: methods and applications

Arnab Sen, Diptiman Sen, K. Sengupta

We present a brief overview of some of the analytic perturbative techniques for the computation of the Floquet Hamiltonian for a periodically driven, or Floquet, quantum many-body system. The key technical points about each of the methods discussed are presented in a pedagogical manner. They are followed by a brief account of some chosen phenomena where these methods have provided useful insights. We provide an extensive discussion of the Floquet-Magnus (FM) expansion, the adiabatic-impulse approximation, and the Floquet perturbation theory. This is followed by a relatively short discourse on the rotating wave approximation, a FM resummation technique and the Hamiltonian flow method. We also provide a discussion of some open problems which may possibly be addressed using these methods.

arXiv:2102.00793•Journal of Physics Condensed Matter•Aug 6, 2021•57 citations

Krylov construction and complexity for driven quantum systems

Amin A. Nizami, Ankit W. Shrestha

Krylov complexity is an important dynamical quantity with relevance to the study of operator growth and quantum chaos and has recently been much studied for various time-independent systems. We initiate the study of K complexity in time-dependent (driven) quantum systems. For periodic time-dependent (Floquet) systems, we develop a natural method for doing the Krylov construction and then define (state and operator) K complexity for such systems. Focusing on kicked systems, in particular the quantum kicked rotor on a torus, we provide a detailed numerical study of the time dependence of Arnoldi coefficients as well as of the K complexity with the system coupling constant interpolating between the weak and strong coupling regimes. We also study the growth of the Krylov subspace dimension as a function of the system coupling constant.

arXiv:2305.00256•Physical review. E•Nov 28, 2023•35 citations

A phase space approach to the wavefunction spreading and operator growth in the Krylov basis

Kunal Pal, Kuntal Pal, Keun-Young Kim

In the Wigner-Weyl phase space formulation of quantum mechanics, we analyse the problem of the spreading of an initial state or an initial operator under time evolution when described in terms of the Krylov basis. After constructing the phase space functions corresponding to the Krylov basis states generated by a Hamiltonian from a given initial state by using the Weyl transformation, we subsequently use them to cast the Krylov state complexity as an integral over the phase space in terms of the Wigner function of the time-evolved initial state, so that the contribution of the classical Liouville equation and higher-order quantum corrections to the Wigner function time evolution equation towards the Krylov state complexity can be identified. Next, we construct the double phase space functions associated with the Krylov basis for operators by using a suitable generalisation of the Weyl transformation applicable for superoperators, and use them to rewrite the Krylov operator complexity as an integral over the double phase space in terms of a generalisation of the usual Wigner function. These results, in particular, show that the complexity measures based on the expansion of a time-evolved state (or an operator) in the Krylov basis can be thought to belong to a general class of complexity measures constructed from the expansion coefficients of the time-dependent Wigner function in an orthonormal basis in the phase space, and help us to connect these complexity measures with measures of complexity of time evolved state based on harmonic expansion of the time-dependent Wigner function.

quant-pharXiv:2601.13872•Journal of High Energy Physics•Jan 21, 2026•1 citation

Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras

András Grabarits, E. Medina-Guerra, Adolfo del Campo

We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place. We establish a direct link between dynamics in the time-dependent Krylov subspace and the underlying Lie-algebraic structure of the Hamiltonian. We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra. In particular, we identify the minimal conditions under which the exact time-dependent Krylov dynamics is naturally determined by the interaction-picture Hamiltonian and governed by an embedded $\mathfrak{sl}(2,\mathbb{C})$ subalgebra. We further show that an exact single-exponential representation of the time-evolution operator gives rise to a distinct time-independent Krylov dynamics in a unitarily related basis, from which the exact time-dependent Krylov dynamics can nevertheless be recovered. We also extend the framework to the oscillator algebra as the simplest extension of the nilpotent Heisenberg--Weyl algebra, and provide further examples, including the translated and dilated harmonic oscillator, systems governed by closed Virasoro subalgebras, a spin in a rotating magnetic field, and higher-dimensional generalizations for multi-level systems. In addition, we introduce a new quantum speed limit to the complexity growth rate generated by a time-dependent generator and show that, for evolutions governed by a Lie algebra, it retains the same functional form as in the time-independent case. Remarkably, saturation of this bound is strongly affected by temporal driving and persists only when the Hamiltonian commutes with itself at different times. These results establish a unified framework for characterizing operator growth and Krylov complexity in time-dependent quantum systems with underlying Lie-algebraic structures.

quant-pharXiv:2605.05290•May 8, 2026•0 citations
We investigate Krylov spread complexity for states evolving under time-dependent Hamiltonians. For periodically driven systems, we formulate the problem within Floquet theory and show how the Magnus expansion provides a systematic approximation when the Floquet Hamiltonian is not available in closed form. We then extend this framework beyond periodic driving and demonstrate that, in addition to the globally truncated Magnus expansion, a piecewise Magnus expansion provides a reliable method when the global expansion loses convergence or accuracy. Our results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems.
The paper studies how to compute state Krylov (spread) complexity under explicitly time-dependent Hamiltonians using effective generators obtained from Floquet theory and Magnus expansions. It treats periodically driven two-level systems and a harmonic oscillator with time-dependent frequency.
For periodic driving, the Krylov basis is constructed from a Floquet Hamiltonian, giving a stroboscopic complexity. For HF=h⋅σH_F=\mathbf h\cdot\boldsymbol\sigmaHF​=h⋅σ and an initial σz\sigma_zσz​ eigenstate, CK(t)=(hx2+hy2)sin⁡2(∣h∣t)/∣h∣2C_K(t)=(h_x^2+h_y^2)\sin^2(|\mathbf h|t)/|\mathbf h|^2CK​(t)=(hx2​+hy2​)sin2(∣h∣t)/∣h∣2. Circular driving is solved exactly in a rotating frame; linear driving is treated near resonance by the rotating-wave approximation and at high frequency by a dressed-frame Magnus expansion. The latter produces a Bessel-renormalized longitudinal field and a second-order transverse correction involving Bessel and Struve functions.
For the oscillator, symplectic propagation determines Bogoliubov coefficients. In the chosen even-Fock Krylov chain, CK(t)=∣ν(t)∣2/2C_K(t)=|\nu(t)|^2/2CK​(t)=∣ν(t)∣2/2. A sudden quench gives an exact oscillatory expression; smooth quenches are computed numerically by composing local Magnus propagators, recovering sudden-quench and approximately adiabatic behavior.
The paper reviews the sufficient convergence condition ∫0t∥H(s)∥ds<π\int_0^t\|H(s)\|ds<\pi∫0t​∥H(s)∥ds<π. Piecewise truncation at order mmm has stated accumulated error O(tΔm)O(t\Delta^m)O(tΔm), assuming controlled stability factors. Limitations include global Magnus convergence and long-time accuracy, accumulated local errors, quasienergy-branch choices, and micromotion outside stroboscopic times. Near zeros of the leading Bessel term, higher Magnus orders determine residual dynamics. Extensions beyond the examples and to operator complexity remain future directions.
We introduce a variational framework for approximating the time-evolution operator U^(t)\hat{U}(t)U^(t) within a physically motivated operator manifold, reformulating quantum dynamics as a tractable problem in operator space using stationary action principle. For periodically driven systems, the resulting approximate evolution operator directly yields an effective Floquet Hamiltonian, offering a non-perturbative alternative to conventional expansion-based methods. The framework is systematically improvable by enlarging the operator pool and naturally incorporates symmetries and physical constraints. When the operator manifold is chosen from the terms of a truncated Magnus expansion, the variational procedure effectively resums the Magnus series within the restricted space, significantly enhancing accuracy. We benchmark the approach on the driven Rabi model, the driven Lipkin-Meshkov-Glick model, and the one-dimensional driven Ising chain, yielding effective Floquet Hamiltonians that are systematically more accurate than low-order Magnus expansions, particularly in regimes where the latter converge poorly, and illustrating applicability to systems with exponentially large Hilbert spaces. Although we focus here on Floquet systems, the formalism applies equally to generic time-dependent Hamiltonians, providing a versatile tool for non-equilibrium quantum dynamics.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.
We present a brief overview of some of the analytic perturbative techniques for the computation of the Floquet Hamiltonian for a periodically driven, or Floquet, quantum many-body system. The key technical points about each of the methods discussed are presented in a pedagogical manner. They are followed by a brief account of some chosen phenomena where these methods have provided useful insights. We provide an extensive discussion of the Floquet-Magnus (FM) expansion, the adiabatic-impulse approximation, and the Floquet perturbation theory. This is followed by a relatively short discourse on the rotating wave approximation, a FM resummation technique and the Hamiltonian flow method. We also provide a discussion of some open problems which may possibly be addressed using these methods.
Krylov complexity is an important dynamical quantity with relevance to the study of operator growth and quantum chaos and has recently been much studied for various time-independent systems. We initiate the study of K complexity in time-dependent (driven) quantum systems. For periodic time-dependent (Floquet) systems, we develop a natural method for doing the Krylov construction and then define (state and operator) K complexity for such systems. Focusing on kicked systems, in particular the quantum kicked rotor on a torus, we provide a detailed numerical study of the time dependence of Arnoldi coefficients as well as of the K complexity with the system coupling constant interpolating between the weak and strong coupling regimes. We also study the growth of the Krylov subspace dimension as a function of the system coupling constant.
In the Wigner-Weyl phase space formulation of quantum mechanics, we analyse the problem of the spreading of an initial state or an initial operator under time evolution when described in terms of the Krylov basis. After constructing the phase space functions corresponding to the Krylov basis states generated by a Hamiltonian from a given initial state by using the Weyl transformation, we subsequently use them to cast the Krylov state complexity as an integral over the phase space in terms of the Wigner function of the time-evolved initial state, so that the contribution of the classical Liouville equation and higher-order quantum corrections to the Wigner function time evolution equation towards the Krylov state complexity can be identified. Next, we construct the double phase space functions associated with the Krylov basis for operators by using a suitable generalisation of the Weyl transformation applicable for superoperators, and use them to rewrite the Krylov operator complexity as an integral over the double phase space in terms of a generalisation of the usual Wigner function. These results, in particular, show that the complexity measures based on the expansion of a time-evolved state (or an operator) in the Krylov basis can be thought to belong to a general class of complexity measures constructed from the expansion coefficients of the time-dependent Wigner function in an orthonormal basis in the phase space, and help us to connect these complexity measures with measures of complexity of time evolved state based on harmonic expansion of the time-dependent Wigner function.
We study quantum dynamics generated by time-dependent Hamiltonians in Krylov space, the minimal subspace in which the evolution takes place. We establish a direct link between dynamics in the time-dependent Krylov subspace and the underlying Lie-algebraic structure of the Hamiltonian. We develop a general framework in which the dynamics in the time-dependent Krylov subspace is generated by ladder operators of the associated Lie algebra. In particular, we identify the minimal conditions under which the exact time-dependent Krylov dynamics is naturally determined by the interaction-picture Hamiltonian and governed by an embedded sl(2,C)\mathfrak{sl}(2,\mathbb{C})sl(2,C) subalgebra. We further show that an exact single-exponential representation of the time-evolution operator gives rise to a distinct time-independent Krylov dynamics in a unitarily related basis, from which the exact time-dependent Krylov dynamics can nevertheless be recovered. We also extend the framework to the oscillator algebra as the simplest extension of the nilpotent Heisenberg--Weyl algebra, and provide further examples, including the translated and dilated harmonic oscillator, systems governed by closed Virasoro subalgebras, a spin in a rotating magnetic field, and higher-dimensional generalizations for multi-level systems. In addition, we introduce a new quantum speed limit to the complexity growth rate generated by a time-dependent generator and show that, for evolutions governed by a Lie algebra, it retains the same functional form as in the time-independent case. Remarkably, saturation of this bound is strongly affected by temporal driving and persists only when the Hamiltonian commutes with itself at different times. These results establish a unified framework for characterizing operator growth and Krylov complexity in time-dependent quantum systems with underlying Lie-algebraic structures.