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Quantum chaos, integrability, and late times in the Krylov basis
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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 suffices 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.
Forward citations
Cited by 7 Pith papers
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Krylov Complexity from Loschmidt Amplitude
Krylov complexity is the ϕ-derivative of a Loschmidt amplitude and is upper-bounded by the volume of the induced (t,ϕ) Fubini-Study geometry.
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Normalized Krylov-Wigner negativity rate matches Krylov variance growth and equals tidal stretch rate R ∝ C P_ρ if and only if Δ=1 in AdS3.
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(A)Symmetric Complexity and the Quantum Mpemba Effect
A new decomposition of Krylov complexity into projected symmetric and asymmetric parts diagnoses the quantum Mpemba effect, but its claimed t=0 predictor is computationally equivalent to time evolution.
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Complexity of PXP scars revisited
In the PXP model, the arch in the Lanczos coefficients is traced to a linear sl(3) part of the Hamiltonian, and the arch width is proposed as a signal distinguishing scarred from thermalizing states.
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The Information Content of Krylov Observables: A Machine Learning Approach
Under chaos, the normalized Wigner negativity χ(t) carries information about the fine return dynamics that spread complexity C(t) cannot, with the asymmetry gap rising from +0.33 to +0.77 across the integrable-to-GUE ...
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Emergence of Krylov complexity through quantum walks: An exploration of the quantum origins of complexity
Reducing a graph walk to distance-layers reproduces Krylov/spread complexity, yielding analytic finite-q SYK Lanczos coefficients and hypercube complexity D sin²(t/D), with faster saturation than classical-walk circuits.
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Quasinormal modes and complexity in saddle-dominated SU(N) spin systems
A family of SU(2) and SU(3) Lipkin-Meshkov-Glick-type Hamiltonians reproduces de Sitter quasinormal-mode densities of states, and late-time probes reveal integrability beneath saddle-dominated scrambling.
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