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Krylov complexity and orthogonal polynomials
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Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution. The construction of that basis relies on the Lanczos algorithm, also known as the recursion method. The mathematics of Krylov complexity can be described in terms of orthogonal polynomials. We provide a pedagogical introduction to the subject and work out analytically a number of examples involving the classical orthogonal polynomials, polynomials of the Hahn class, and the Tricomi-Carlitz polynomials.
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Cited by 12 Pith papers
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Comments on holographic spread complexity
The momentum–spread-complexity relation requires generalized coherent states adapted to the spacetime symmetry algebra; semiclassical spreading alone cannot produce a momentum–complexity correspondence.
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Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity
Polynomial changes of the initial state in Krylov complexity are solved exactly via Christoffel transforms of the spectral measure, yielding finite-band amplitude transfer and projected-kernel complexity formulas with...
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Streamlined Krylov construction and classification of ergodic Floquet systems
A Szegő/CMV Krylov construction maps Floquet unitary dynamics to a five-diagonal chain, with a conjectured classification of chaos and integrability by Verblunsky coefficient asymptotics.
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Recursion Coefficients and Krylov Dynamics in Polynomial Random Matrix Models
Recursion coefficients for high-degree asymmetric polynomial random matrix models are computed efficiently via a moment recursion, with large-n asymptotics reproducing Freud's conjecture and transition regions mapped ...
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Spectral Topology and Universal Krylov Dynamics
Spectral topology, meaning the gap structure of the support, refines the universal classification of Krylov operator growth: gaps produce quasiperiodic Lanczos coefficients, and a gap closing with a double density zer...
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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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Black holes from chaos
Chaotic thermal correlators can be computed by mapping Lanczos coefficients to a black-hole scattering problem, giving quasinormal-mode resonances from short-time data.
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Spread complexity and the saturation of wormhole size
For finite-N DSSYK, the chord basis is the early part of the physical Krylov basis, and spread complexity, taken as the non-perturbative ER bridge size, saturates after a universality-class-dependent peak and slope.
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Krylov Complexity in the Schr\"odinger Field Theory
For bosonic and fermionic Schrödinger fields with chemical potential μ≤0, the Lanczos coefficients grow linearly and the Krylov complexity grows exponentially with an extracted rate near 2.75/β, below the 4/β slope pr...
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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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Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information
Time-averaged quantum Fisher information in Krylov space changes slope at the PT transition (gamma=1) and saturates near the entanglement transition (gamma=2) in the monitored SSH model, suggesting it as a probe of both.
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Statistics and Complexity of Wavefunction Spreading in Quantum Dynamical Systems
The moments of the spreading-operator measurement distribution are generalized spread complexities, which for GUE Hamiltonians peak more sharply at higher order and obey a norm bound.
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