Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas

• URL: http://arxiv.org/abs/2603.19359v1
• Date: Thu, 19 Mar 2026 18:00:07 GMT
• Title: Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas
• Abstract summary: logarithmic Krylov (logK) complexity is a probe of early-time operator scrambling without false positives.<n>We show that logK-complexity discriminates between genuine and saddle-dominated scrambling at early times.<n>We extend the Krylov formalism in classical dynamical systems to define classical versions of these operator growth measures.
• Score: 3.3587645077393655
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose and test logarithmic Krylov (logK) complexity, an operator growth measure akin to Krylov complexity defined through a replica approach, as a viable probe of early-time operator scrambling without false positives. In finite-dimensional quantum systems, such as the Lipkin--Meshkov--Glick (LMG) model and the mixed-field Ising model at the chaotic point, we provide numerical evidence that logK-complexity discriminates between genuine and saddle-dominated scrambling at early times, correctly avoiding the exponential contribution coming from the unstable saddle in the former case, and closely tracking the conventional Krylov complexity in the latter. In integrable quantum systems admitting infinite-dimensional Krylov subspaces, such as the SYK$_{2}$ model and the quantum inverted harmonic oscillator, we show that by modifying the Krylov spreading operator, obtained through generalizing the analytic continuation procedure in the replica trick, the logK complexity can be refined to capture the integrable properties of the theories. We supplement these analyses by extending the Krylov formalism in classical dynamical systems and defining classical versions of these operator growth measures, showing that the false positives arising from unstable saddles in classical phase space are non-existent.

Related papers

• From phase space to Krylov space, one shell at a time [0.0]
  We develop and study the classical Lanczos algorithm allowing us to define Krylov complexity.<n>We show, using general methods of quantum mechanics in phase space, that the $hbar to 0$ limit of the usual quantum mechanical Krylov framework smoothly goes over into the classical one.<n>In theories with well-defined semiclassical limits, we show that classical Krylov complexity accurately approximates quantum complexity at early enough times.
  arXiv  Detail & Related papers  (2026-07-14T09:54:05Z)
• Entanglement-facilitated macroscopic cluster formation in quantum many-body dynamics [42.89492957665636]
  We find that initial-state entanglement suppresses the proliferation of true-vacuum bubbles and stabilises macroscopic connected clusters.<n>Our results establish a connection between initial-state preparation and the preservation of global structures.
  arXiv  Detail & Related papers  (2026-05-21T18:21:14Z)
• A rigorous quasipolynomial-time classical algorithm for SYK thermal expectations [51.660331450043806]
  Estimating local observables in Gibbs states is a central problem in quantum simulation.<n>We give a proof of a quasipolynomial-time classical algorithm that estimates SYK local thermal expectations at sufficiently high constant temperature.<n>Our result introduces a new Wick-pair expansion that we expect to be broadly useful for quantum many-body systems.
  arXiv  Detail & Related papers  (2026-04-22T21:14:04Z)
• A Schwinger-Keldysh Formulation of Semiclassical Operator Dynamics [0.0]
  We develop a real-time Schwinger-Keldysh formulation of Krylov dynamics that treats Krylov complexity as an in-in observable generated by a closed time contour path integral path.<n>The resulting functional exposes an emergent phase-space description in which the Lanczos coefficients define an effective Hamiltonian governing operator motion along the Krylov chain.<n>This formulation reorganises Krylov complexity into a dynamical field-theoretic framework and identifies new fluctuation diagnostics of operator growth in closed quantum systems.
  arXiv  Detail & Related papers  (2026-02-02T13:51:01Z)
• Semiclassical entanglement entropy for spin-field interaction [41.99844472131922]
  We study a general bipartite quantum system consisting of a spin interacting with a bosonic field.<n>Our goal is to develop a semiclassical framework to describe the entanglement dynamics between these two subsystems.
  arXiv  Detail & Related papers  (2026-01-22T14:07:56Z)
• Sample-Based Krylov Quantum Diagonalization for the Schwinger Model on Trapped-Ion and Superconducting Quantum Processors [26.315169722500556]
  We apply the recently proposed Sample-based Krylov Quantum Diagonalization (SKQD) method to lattice gauge theories.<n>We study the dependence of the ground-state energy and particle number on the value of the $theta$-term, accurately capturing the model's phase structure.<n>We show that SKQD substantially reduces the effective Hilbert space, and although the Krylov space dimension still scales exponentially, the slower growth underscores its promise for simulating lattice gauge theories in larger volumes.
  arXiv  Detail & Related papers  (2025-10-30T19:21:06Z)
• Fast-Forward Lattice Boltzmann: Learning Kinetic Behaviour with Physics-Informed Neural Operators [37.65214107289304]
  We introduce a physics-informed neural operator framework for the lattice Boltzmann equation (LBE)<n>Our framework is discretization-invariant, enabling models trained on coarse lattices to generalise to finer ones.<n>Results demonstrate robustness across complex flow scenarios, including von Karman vortex shedding, ligament breakup, and bubble adhesion.
  arXiv  Detail & Related papers  (2025-09-26T14:36:23Z)
• Krylov Complexity for Open Quantum System: Dissipation and Decoherence [0.14337588659482522]
  We investigate Krylov complexity in open quantum systems using Lindblad master equations for bosonic bath models.<n>We find that Krylov complexity saturates in the full system and reproduces the expected dissipative behavior when the decoherence term is suppressed.<n>However, Krylov complexity appears insensitive to the onset of decoherence, as no clear distinctive signature is observed.
  arXiv  Detail & Related papers  (2025-09-18T10:12:01Z)
• Operator K-complexity in DSSYK: Krylov complexity equals bulk length [0.0]
  We study the notion of complexity under time evolution in chaotic quantum systems with holographic duals.<n>We find that Krylov complexity is given by the expectation value of a length operator acting on the Hilbert space of the theory.<n>We conclude that evolution on the Krylov chain can equivalently be understood as a particle moving in a Morse potential.
  arXiv  Detail & Related papers  (2024-12-19T18:54:30Z)
• KPZ scaling from the Krylov space [83.88591755871734]
  Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang scaling in late-time correlators and autocorrelators has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis.
  arXiv  Detail & Related papers  (2024-06-04T20:57:59Z)
• Operator dynamics in Lindbladian SYK: a Krylov complexity perspective [0.0]
  We analytically establish the linear growth of two sets of coefficients for any generic jump operators. We find that the Krylov complexity saturates inversely with the dissipation strength, while the dissipative timescale grows logarithmically.
  arXiv  Detail & Related papers  (2023-11-01T18:00:06Z)
• Work statistics, quantum signatures and enhanced work extraction in quadratic fermionic models [62.997667081978825]
  In quadratic fermionic models we determine a quantum correction to the work statistics after a sudden and a time-dependent driving. Such a correction lies in the non-commutativity of the initial quantum state and the time-dependent Hamiltonian. Thanks to the latter, one can assess the onset of non-classical signatures in the KDQ distribution of work.
  arXiv  Detail & Related papers  (2023-02-27T13:42:40Z)
• Krylov Localization and suppression of complexity [0.0]
  We investigate Krylov complexity for the case of interacting integrable models at finite size. We find that complexity saturation is suppressed as compared to chaotic systems. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain.
  arXiv  Detail & Related papers  (2021-12-22T18:45:32Z)
• Entanglement dynamics of spins using a few complex trajectories [77.34726150561087]
  We consider two spins initially prepared in a product of coherent states and study their entanglement dynamics. We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator.
  arXiv  Detail & Related papers  (2021-08-13T01:44:24Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.