Changing the initial state to Q(H)|K0⟩ is exactly a Christoffel reweighting of the spectral measure by |Q|²; Krylov complexity then transfers from the reference problem through finite-band connectors and finite-rank kernel projections, with closed forms in Charlier, Krawtchouk and Chebyshev chains.
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Quantum chaos and the complexity of spread of states
Canonical reference. 80% of citing Pith papers cite this work as background.
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abstract
We propose a measure of quantum state complexity defined by minimizing the spread of the wave-function over all choices of basis. Our measure is controlled by the "survival amplitude" for a state to remain unchanged, and can be efficiently computed in theories with discrete spectra. For continuous Hamiltonian evolution, it generalizes Krylov operator complexity to quantum states. We apply our methods to the harmonic and inverted oscillators, particles on group manifolds, the Schwarzian theory, the SYK model, and random matrix models. For time-evolved thermofield double states in chaotic systems our measure shows four regimes: a linear "ramp" up to a "peak" that is exponential in the entropy, followed by a "slope" down to a "plateau". These regimes arise in the same physics producing the slope-dip-ramp-plateau structure of the Spectral Form Factor. Specifically, the complexity slope arises from spectral rigidity, distinguishing different random matrix ensembles.
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representative citing papers
Defines tripartite complexity and complexity gap for three-subsystem states and reports that the gap has definite sign across holographic CV, Fisher-Rao, and Krylov measures, suggesting it as a building block for complexity inequalities.
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.
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.
Holographic complexity measures show universal linear growth followed by late-time saturation, proven necessary and sufficient via pole structures in the energy basis using the residue theorem, arising from random matrix statistics.
Wigner negativity in Krylov space stays O(1) or grows as t^{1/2} (without Hilbert-space scaling) in 2d CFTs, one-cut matrix models, and double-scaled SYK, indicating emergent semiclassicality.
D0-branes in ABJM, rotating D3-branes, and wound strings realize holographic spread complexity via proper momentum and Routhian prescriptions that match short-time Krylov behavior.
Orbifolds of N=4 SYM produce SCFTs whose dilatation operator in a subsector is realized by a tunable spin chain whose eigenvalue statistics exhibit chaos for specific marginal couplings.
Krylov complexity's Taylor coefficients recursively determine all Lanczos coefficients, making it a complete descriptor of operator dynamics, with caveats for spread complexity.
Exact Krylov correlators in sl(2,R) models are proportional to radial momenta in BTZ black holes, extending the complexity-momentum correspondence to include fluctuations.
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.
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.
Protected and few-body sectors in N=4 SYM exhibit integrable Krylov dynamics with a_n=2Mg and b_n→Mg, insufficient for testing gravitational universality of complexity growth; a finite-density program is proposed to test dependence only on coarse thermodynamic data.
In SdS black hole holography, CV and CV2.0 complexities grow linearly while CA growth vanishes due to finite action, with matching rates between static patch and dS/CFT schemes.
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 timelike quantum focusing conjecture implies a complexity-based quantum strong energy condition and a complexity bound analogous to the covariant entropy bound for suitable codimension-0 field theory complexity measures.
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.
citing papers explorer
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Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity
Changing the initial state to Q(H)|K0⟩ is exactly a Christoffel reweighting of the spectral measure by |Q|²; Krylov complexity then transfers from the reference problem through finite-band connectors and finite-rank kernel projections, with closed forms in Charlier, Krawtchouk and Chebyshev chains.
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Complexity Inequalities for Quantum Subsystems
Defines tripartite complexity and complexity gap for three-subsystem states and reports that the gap has definite sign across holographic CV, Fisher-Rao, and Krylov measures, suggesting it as a building block for complexity inequalities.
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Holographic Krylov Complexity for Charged, Composite and Extended Probes
Holographic Krylov complexity for charged composite and extended probes retains universal leading large-time growth but acquires structure-dependent subleading corrections.
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Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas
LogK complexity via replicas distinguishes genuine scrambling from saddle effects in quantum and classical systems and refines the measure for integrable cases.
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Krylov Subspace Dynamics as Near-Horizon AdS$_2$ Holography
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.
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Universal Time Evolution of Holographic and Quantum Complexity
Holographic complexity measures show universal linear growth followed by late-time saturation, proven necessary and sufficient via pole structures in the energy basis using the residue theorem, arising from random matrix statistics.
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Wigner negativity in Krylov space and emergent semiclassicality
Wigner negativity in Krylov space stays O(1) or grows as t^{1/2} (without Hilbert-space scaling) in 2d CFTs, one-cut matrix models, and double-scaled SYK, indicating emergent semiclassicality.
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Holographic Spread Complexity from Branes and Strings
D0-branes in ABJM, rotating D3-branes, and wound strings realize holographic spread complexity via proper momentum and Routhian prescriptions that match short-time Krylov behavior.
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Controlled Chaos in 4D SCFTs
Orbifolds of N=4 SYM produce SCFTs whose dilatation operator in a subsector is realized by a tunable spin chain whose eigenvalue statistics exhibit chaos for specific marginal couplings.
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Krylov complexity has it all
Krylov complexity's Taylor coefficients recursively determine all Lanczos coefficients, making it a complete descriptor of operator dynamics, with caveats for spread complexity.
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Krylov Correlators in $\mathfrak{sl}(2,\mathbb R)$ Models: Exact Results and Holographic Complexity
Exact Krylov correlators in sl(2,R) models are proportional to radial momenta in BTZ black holes, extending the complexity-momentum correspondence to include fluctuations.
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Bridging Krylov Complexity and Universal Analog Quantum Simulator
Generalized Krylov complexity predicts the minimum time to realize target operations in analog quantum simulators such as Rydberg atom arrays.
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Cosmological brick walls & quantum chaotic dynamics of de Sitter horizons
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.
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Cosmological Entanglement Entropy from the von Neumann Algebra of Double-Scaled SYK & Its Connection with Krylov Complexity
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.
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Toward Krylov-based holography in double-scaled SYK
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.
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On the Universality of Probe Complexity in $\mathcal{N}=4$ SYM
Protected and few-body sectors in N=4 SYM exhibit integrable Krylov dynamics with a_n=2Mg and b_n→Mg, insufficient for testing gravitational universality of complexity growth; a finite-density program is proposed to test dependence only on coarse thermodynamic data.
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Holographic complexity of de-Sitter black holes
In SdS black hole holography, CV and CV2.0 complexities grow linearly while CA growth vanishes due to finite action, with matching rates between static patch and dS/CFT schemes.
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Krylov complexity from a simple quantum mechanical model for a radiating black hole
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.
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A Timelike Quantum Focusing Conjecture
A timelike quantum focusing conjecture implies a complexity-based quantum strong energy condition and a complexity bound analogous to the covariant entropy bound for suitable codimension-0 field theory complexity measures.
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Probing the Chaos to Integrability Transition in Double-Scaled SYK
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.
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Krylov Complexity for Open Quantum System: Dissipation and Decoherence
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
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Complexity of Quadratic Quantum Chaos
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.
- q-Askey Deformations of Double-Scaled SYK
- Deforming the Double-Scaled SYK & Reaching the Stretched Horizon From Finite Cutoff Holography