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arXiv:2607.10921

Krylov Complexity from Loschmidt Amplitude

Debarghya Chakraborty

hep-thcond-mat.stat-mechquant-phSubmitted Jul 12, 2026• Announced Jul 14, 2026• 0 citations• Revised Sep 28, 2026 (v2)
v1 · Jul 12, 2026v2 · Sep 28, 2026
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Abstract

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.

Comments
Several typos corrected and references added. The discussion for late-time complexity distribution for bounded spectral function has been expanded and moved to Section III.C. 36 pages, 7 figures

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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.
Abstract Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace KM(H,η)\mathcal {K}_M(\mathcal {H},\eta )KM​(H,η) spanned by the multiple applications of the Liouville operator L\mathcal {L}L defined by the commutator in terms of a Hamiltonian H\mathcal {H}H, L:=[H,⋅]\mathcal {L}:=[\mathcal {H},\cdot ]L:=[H,⋅] acting on an operator η, KM(H,η)=span{η,Lη,…,LM−1η}\mathcal {K}_M(\mathcal {H},\eta )=\text{span}\lbrace \eta ,\mathcal {L}\eta ,\ldots ,\mathcal {L}^{M-1}\eta \rbraceKM​(H,η)=span{η,Lη,…,LM−1η}. For a given inner product ( ·, ·) of the operators, the orthonormal basis {On}\lbrace \mathcal {O}_n\rbrace{On​} is constructed from O0=η/(η,η)\mathcal {O}_0=\eta /\sqrt{(\eta ,\eta )}O0​=η/(η,η)​ by Lanczos algorithm. The moments μm=(O0,LmO0)\mu _m=(\mathcal {O}_0,\mathcal {L}^m\mathcal {O}_0)μm​=(O0​,LmO0​) are closely related to the important data {bn} called Lanczos coefficients. I present the exact and explicit expressions of the moments {μm} for 16 quantum mechanical systems which are exactly solvable both in the Schrödinger and Heisenberg pictures. The operator η is the variable of the eigenpolynomials. Among them six systems show a clear sign of ‘non-complexity’ as vanishing higher Lanczos coefficients bm = 0, m ≥ 3.
Krylov complexity has recently been proposed as a quantum probe of chaos. The Krylov exponent characterizing the exponential growth of Krylov complexity is conjectured to upper-bound the Lyapunov exponent. We compute the Krylov and the Lyapunov exponents in the Sachdev-Ye-Kitaev model and in some of its deformations. We do this analysis both at infinite and finite temperatures, in models where the number of fermionic interactions is both finite and infinite. We consider deformations that interpolate between two regions of near-maximal chaos and deformations that become nearly integrable at low temperatures. In all cases, we find that the Krylov exponent upper-bounds the Lyapunov one. However, we find that while the Lyapunov exponent can have nonmonotonic behavior as a function of temperature, in all studied examples the Krylov exponent behaves monotonically. For instance, we find models where the Lyapunov exponent goes to zero at low temperatures, while the Krylov exponent saturates to its maximal bound. We speculate on the possibility that this monotonicity might be a generic feature of the Krylov exponent in quantum systems evolving under unitary evolution.
In closed quantum systems, Krylov complexity admits a geometric description; operator growth is equivalent to Hamiltonian flow in an emergent phase space whose structure is fixed by the Lanczos coefficients. We show that this picture survives, albeit in a fundamentally altered form, once the system is coupled to an environment.Using a Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, we derive an effective action for operator growth under Lindblad dynamics. Even for the minimal case of dephasing, the phase-space dynamics ceases to be Hamiltonian; environmental coupling generates diffusion in the variable conjugate to Krylov depth, converting deterministic trajectories in to stochastic ones. The hyperbolic mechanism underlying exponential complexity growth is therefore broadened and, beyond a parametrically controlled scale, destroyed.This identifies dissipation as a relevant perturbation of the chaotic Krylov fixed point and reveals operator growth in open systems as a problem of stochastic dynamics in an emergent phase space.
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.
This paper establishes that Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density. To demonstrate this equivalence, an explicit recursive algorithm is constructed to calculate Lanczos coefficients from the Taylor expansion of the Krylov complexity around t=0t=0t=0. Furthermore, the paper discusses the distinction between Krylov and spread complexity, clarifying why a similar recursive algorithm cannot exist for the latter without additional dynamical input. These results provide a ``proof of principle'' for using Krylov complexity as a complete characterization of operator evolution in quantum systems.
Abstract Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.