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:2402.09522 · Journal of High Energy Physics

Krylov complexity of density matrix operators

Paweł Caputa, Hyun-Sik Jeong, Sinong Liu, Juan F. Pedraza, Le-Chen Qu

Published May 31, 2024• 59 citations
arXivHTMLPDF

Abstract

A bstract Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ( C K ) and Spread complexity ( C S ) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators . After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between C K and 2 C S . Furthermore, for maximally entangled pure states, we find that the moment-generating function of C K becomes the Spectral Form Factor and, at late-times, C K is simply related to NC S for N ≥ 2 within the N -dimensional Hilbert space. Notably, we confirm that C K = 2 C S holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.

DOI
10.1007/jhep05(2024)337

Related papers

Statistics and complexity of wavefunction spreading in quantum dynamical systems

Y. W. Fu, Keun-young Kim A, Kunal Pal +1 more

A bstract We consider the statistics of the results of a measurement of the spreading operator in the Krylov basis generated by the Hamiltonian of a quantum system starting from a specified initial pure state. We first obtain the probability distribution of the results of measurements of this spreading operator at a certain instant of time, and compute the characteristic function of this distribution. We show that the moments of this characteristic function are related to the so-called generalised spread complexities, and obtain expressions for them in several cases when the Hamiltonian is an element of a Lie algebra. Furthermore, by considering a continuum limit of the Krylov basis, we show that the generalised spread complexities of higher orders have a peak in the time evolution for a random matrix Hamiltonian belonging to the Gaussian unitary ensemble. We also obtain an upper bound in the change in generalised spread complexity at an arbitrary time in terms of the operator norm of the Hamiltonian and discuss the significance of these results.

arXiv:2411.09390•Journal of High Energy Physics•Jun 13, 2025•10 citations

Krylov complexity in the Schrödinger field theory

Peng-Zhang He, Hai-Qing Zhang

A bstract We investigate the Krylov complexity of Schrödinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble which includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator [ H , ·] on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients { b n } are almost independent of the chemical potential, while { a n } are dependent on the chemical potential. Both { a n } and { b n } show linear relationships with respect to n . For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions | φ 0 ( t )| 2 . The Krylov complexity grows exponentially with time, but its asymptotic scaling factor λ K is significantly smaller than the twice of the slope of the { b n } coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of { b n }.

arXiv:2411.16302•Journal of High Energy Physics•Mar 19, 2025•4 citations

Higher-order Krylov state complexity in random matrix quenches

Hugo A. Camargo, Yichao Fu, Viktor Jahnke +2 more

A bstract In quantum many-body systems, time-evolved states typically remain confined to a smaller region of the Hilbert space known as the Krylov subspace . The time evolution can be mapped onto a one-dimensional problem of a particle moving on a chain, where the average position 〈 n 〉 defines Krylov state complexity or spread complexity. Generalized spread complexities, associated with higher-order moments 〈 n p 〉 for p > 1, provide finer insights into the dynamics. We investigate the time evolution of generalized spread complexities following a quantum quench in random matrix theory. The quench is implemented by transitioning from an initial random Hamiltonian to a post-quench Hamiltonian obtained by dividing it into four blocks and flipping the sign of the off-diagonal blocks. This setup captures universal features of chaotic quantum quenches. When the initial state is the thermofield double state of the post-quench Hamiltonian, a peak in spread complexity preceding equilibration signals level repulsion, a hallmark of quantum chaos. We examine the robustness of this peak for other initial states, such as the ground state or the thermofield double state of the pre-quench Hamiltonian. To quantify this behavior, we introduce a measure based on the peak height relative to the late-time saturation value. In the continuous limit, higher-order complexities show increased sensitivity to the peak, supported by numerical simulations for finite-size random matrices.

arXiv:2412.16472•Journal of High Energy Physics•Jul 16, 2025•11 citations

Krylov complexity and spectral density of BMN matrix model

Dibakar Roychowdhury

We use Krylov complexity as a typical diagnostic of quantum many body dynamics in the context of BMN matrix model at large mass gap. We calculate several physical entities, for example, the moments and return amplitudes at large mass deformation. We calculate them for both spread complexity of states as well as Krylov growth of operators in the matrix model. We propose a general expression for the moments at any given order $n$. We also discuss orthogonal polynomials for spread as well as operator Krylov complexity. In the context of the Krylov operator growth we further compute the spectral function and the density of states. A careful analysis of the spectral function near the resonance reveals various IR divergences in addition to the physical collective modes. These collective modes appear to be exceptionally stable due to large mass gap. On the other hand, the UV of the theory appears to be extremely damped due to higher scattering rates. We carefully diagnose these IR divergences near resonance which reveals that these collective modes are in fact non-propagating and should be thought of as diffusion modes or relaxation modes with infinite relaxation time. We also calculate the Krylov variance and Krylov entropy for the matrix model and in particular at early time. The linear early time growth of the Krylov entropy confirms the onset of quantum chaos in the matrix model at large mass gap.

hep-thCross-listedarXiv:2607.24632•Jul 28, 2026•0 citations

Quantum chaos, integrability, and late times in the Krylov basis

Vijay Balasubramanian, Javier M. Magán, Qingyue Wu

Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by random matrix theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffice to produce the full long-time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long-time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra.

arXiv:2312.03848•Physical review. E•Jan 15, 2025•25 citations

Krylov Complexity from Loschmidt Amplitude

Debarghya Chakraborty

Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable $φ$. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and $φ$ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the $φ$-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.

hep-thCross-listedarXiv:2607.10921•Jul 14, 2026•0 citations
A bstract Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ( C K ) and Spread complexity ( C S ) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators . After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between C K and 2 C S . Furthermore, for maximally entangled pure states, we find that the moment-generating function of C K becomes the Spectral Form Factor and, at late-times, C K is simply related to NC S for N ≥ 2 within the N -dimensional Hilbert space. Notably, we confirm that C K = 2 C S holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.
A bstract We consider the statistics of the results of a measurement of the spreading operator in the Krylov basis generated by the Hamiltonian of a quantum system starting from a specified initial pure state. We first obtain the probability distribution of the results of measurements of this spreading operator at a certain instant of time, and compute the characteristic function of this distribution. We show that the moments of this characteristic function are related to the so-called generalised spread complexities, and obtain expressions for them in several cases when the Hamiltonian is an element of a Lie algebra. Furthermore, by considering a continuum limit of the Krylov basis, we show that the generalised spread complexities of higher orders have a peak in the time evolution for a random matrix Hamiltonian belonging to the Gaussian unitary ensemble. We also obtain an upper bound in the change in generalised spread complexity at an arbitrary time in terms of the operator norm of the Hamiltonian and discuss the significance of these results.
A bstract We investigate the Krylov complexity of Schrödinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble which includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator [ H , ·] on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients { b n } are almost independent of the chemical potential, while { a n } are dependent on the chemical potential. Both { a n } and { b n } show linear relationships with respect to n . For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions | φ 0 ( t )| 2 . The Krylov complexity grows exponentially with time, but its asymptotic scaling factor λ K is significantly smaller than the twice of the slope of the { b n } coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of { b n }.
A bstract In quantum many-body systems, time-evolved states typically remain confined to a smaller region of the Hilbert space known as the Krylov subspace . The time evolution can be mapped onto a one-dimensional problem of a particle moving on a chain, where the average position 〈 n 〉 defines Krylov state complexity or spread complexity. Generalized spread complexities, associated with higher-order moments 〈 n p 〉 for p > 1, provide finer insights into the dynamics. We investigate the time evolution of generalized spread complexities following a quantum quench in random matrix theory. The quench is implemented by transitioning from an initial random Hamiltonian to a post-quench Hamiltonian obtained by dividing it into four blocks and flipping the sign of the off-diagonal blocks. This setup captures universal features of chaotic quantum quenches. When the initial state is the thermofield double state of the post-quench Hamiltonian, a peak in spread complexity preceding equilibration signals level repulsion, a hallmark of quantum chaos. We examine the robustness of this peak for other initial states, such as the ground state or the thermofield double state of the pre-quench Hamiltonian. To quantify this behavior, we introduce a measure based on the peak height relative to the late-time saturation value. In the continuous limit, higher-order complexities show increased sensitivity to the peak, supported by numerical simulations for finite-size random matrices.
We use Krylov complexity as a typical diagnostic of quantum many body dynamics in the context of BMN matrix model at large mass gap. We calculate several physical entities, for example, the moments and return amplitudes at large mass deformation. We calculate them for both spread complexity of states as well as Krylov growth of operators in the matrix model. We propose a general expression for the moments at any given order nnn. We also discuss orthogonal polynomials for spread as well as operator Krylov complexity. In the context of the Krylov operator growth we further compute the spectral function and the density of states. A careful analysis of the spectral function near the resonance reveals various IR divergences in addition to the physical collective modes. These collective modes appear to be exceptionally stable due to large mass gap. On the other hand, the UV of the theory appears to be extremely damped due to higher scattering rates. We carefully diagnose these IR divergences near resonance which reveals that these collective modes are in fact non-propagating and should be thought of as diffusion modes or relaxation modes with infinite relaxation time. We also calculate the Krylov variance and Krylov entropy for the matrix model and in particular at early time. The linear early time growth of the Krylov entropy confirms the onset of quantum chaos in the matrix model at large mass gap.
Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by random matrix theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffice to produce the full long-time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long-time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra.
Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable φφφ. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and φφφ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the φφφ-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.