In open quantum systems, environmental coupling turns deterministic Krylov phase-space trajectories into stochastic ones by adding diffusion, destroying the hyperbolic mechanism for exponential complexity growth beyond a controlled scale.
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A Universal Operator Growth Hypothesis
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Holographic Krylov complexity for charged composite and extended probes retains universal leading large-time growth but acquires structure-dependent subleading corrections.
LogK complexity via replicas distinguishes genuine scrambling from saddle effects in quantum and classical systems and refines the measure for integrable cases.
A Pauli-transfer-matrix analysis of QELMs reveals the full set of nonlinear Pauli features generated by encoding and transformed by quantum channels, producing an interpretable classical nonlinear vector autoregression model that approximates flow maps in dynamical systems.
In the continuum limit the discrete Krylov chain becomes a Klein-Gordon field in AdS2, with Lanczos growth rate α identified as πT, recovering the maximal chaos bound and requiring the Breitenlohner-Freedman bound for consistency.
Photon rings around black holes saturate the quantum chaos bound via Lyapunov exponents of null geodesics and OTOCs in the near-ring region.
SYK disorder is shown to be an approximate unitary k-design for poly(N) k; under the planted-SYK hardness conjecture this yields gravitationally pseudorandom unitaries, implying cryptographic censorship in JT gravity with the regularized maximal geodesic length as distinguisher.
Generalized Krylov complexity predicts the minimum time to realize target operations in analog quantum simulators such as Rydberg atom arrays.
Brick-wall spectra in de Sitter space show long-range chaotic signatures via spectral form factor and Krylov complexity even when conventional level repulsion is absent.
In holographic 6d N=(1,0) SCFTs, generalized proper momentum of infalling particles grows linearly at late times, with early dynamics modified by SU(2)_R charge and quiver spreading.
Deformations of the double-scaled SYK model via finite-cutoff holography produce Krylov complexity as wormhole length and realize Susskind's stretched horizon proposal through targeted T² deformations in the high-energy spectrum.
Algebraic entanglement entropy from type II1 algebras in double-scaled SYK is matched via triple-scaling limits to Ryu-Takayanagi areas in (A)dS2, reproducing Bekenstein-Hawking and Gibbons-Hawking formulas for specific regions while depending on Krylov complexity of the Hartle-Hawking state.
Establishes a threefold duality linking Krylov complexity growth rate to wormhole velocity and proper momentum in DSSYK holography, with higher moments capturing replica wormholes and Krylov entropy equaling parent-geometry von Neumann entropy after tracing baby universes.
Spectral functions of SYK, p-spin, and SU(M) Heisenberg models show exponential tails in spin-glass phases and quasiparticle families in spin-liquid phases, with a proof that exponential decay blocks detection of bulk causal structure.
Krylov complexity is equivalent to Lanczos coefficients, return amplitude, and spectral density for operator dynamics, via an explicit recursive algorithm from its t=0 Taylor expansion.
A simplified mini-BMN matrix model for a radiating black hole exhibits early-time chaotic growth of Krylov complexity followed by late-time saturation to a plateau consistent with equilibration.
A first-order phase transition in the Berkooz-Brukner-Jia-Mamroud interpolating model causes chord number, Krylov complexity, and operator size to switch discontinuously from chaotic (linear/exponential) to quasi-integrable (quadratic) growth.
Krylov complexity saturates in the full high-temperature Caldeira-Leggett system, reproduces dissipative features when decoherence is suppressed, shows oscillations when dissipation is suppressed, and remains insensitive to decoherence onset because the Krylov basis differs from the conventional one
Hard-core boson two-body models with random interactions exhibit chaotic spectral statistics, operator growth, and eigenstate properties approaching those of random matrices and the SYK model.
Krylov complexity equals Fubini-Study volume for closed and open two-mode squeezed states, providing analytic support for the generalized CV conjecture via information geometry.
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