Krylov complexity and orthogonal polynomials
arXiv:2205.12815 · doi:10.1016/j.nuclphysb.2022.115948
Abstract
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.
arXiv comment78 pages, 9 figures, comments welcome
References in corpus (32)
- Chaos and Quantum Thermalization
- Thermalization and its mechanism for generic isolated quantum systems
- From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
- A bound on chaos
- Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
- Comments on the Sachdev-Ye-Kitaev model
- Black holes and the butterfly effect
- Black holes as mirrors: quantum information in random subsystems
- Ultimate physical limits to computation
- Fast Scramblers
- Complexity, action, and black holes
- Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
- Quantum Computation as Geometry
- Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
- Quantum chaos and the complexity of spread of states
- Operator complexity: a journey to the edge of Krylov space
- On The Evolution Of Operator Complexity Beyond Scrambling
- Krylov complexity in conformal field theory
- Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK
- Toda chain flow in Krylov space
- Krylov Localization and suppression of complexity
- Quantum complexity and topological phases of matter
- Random Matrix Theory for Complexity Growth and Black Hole Interiors
- Krylov Complexity in Quantum Field Theory
- A statistical mechanism for operator growth
- Universal relation for operator complexity
- Pedagogical introduction to SYK model and 2D Dilaton Gravity
- On operator growth and emergent Poincaré symmetries
- Three Lectures on Complexity and Black Holes
- Chaotic Fast Scrambling At Black Holes
- Orthogonal polynomials, a short introduction
- The geometry of quantum computation
Cited by in corpus (51)
- Operator growth and Krylov construction in dissipative open quantum systems
- Universal chaotic dynamics from Krylov space
- Quantum Dynamics in Krylov Space: Methods and Applications
- Krylov Complexity in Open Quantum Systems
- Krylov Complexity in Free and Interacting Scalar Field Theories with Bounded Power Spectrum
- Krylov complexity and chaos in quantum mechanics
- Spread Complexity and Topological Transitions in the Kitaev Chain
- Krylov complexity in quantum field theory, and beyond
- Krylov complexity in large- and double-scaled SYK model
- Krylov complexity of density matrix operators
- Spectral and Krylov Complexity in Billiard Systems
- A universal approach to Krylov State and Operator complexities
- Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model
- Spread complexity in saddle-dominated scrambling
- From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators
- Quantum complexity in gravity, quantum field theory, and quantum information science
- State Dependence of Krylov Complexity in CFTs
- Quantum state complexity meets many-body scars
- Operator growth and Krylov Complexity in Bose-Hubbard Model
- Krylov Complexity in Lifshitz-type Scalar Field Theories
- Krylov Complexity and Spectral Form Factor for Noisy Random Matrix Models
- Entanglement and geometry from subalgebras of the Virasoro algebra
- Spread complexity for measurement-induced non-unitary dynamics and Zeno effect
- Krylov Complexity and Dynamical Phase Transition in the quenched LMG model
- Krylov complexity and Trotter transitions in unitary circuit dynamics
- Inflationary Krylov complexity
- Tuning the Quantum Mpemba Effect in Isolated System by Initial State Engineering
- Universal Hypothesis of Autocorrelation Function from Krylov Complexity
- Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information
- Measurable Krylov Spaces and Eigenenergy Count in Quantum State Dynamics
- Black Holes and Marchenko-Pastur Distribution
- Exactly solvable models for universal operator growth
- Operator size growth in Lindbladian SYK
- Krylov Complexity of Fermionic and Bosonic Gaussian States
- Streamlined Krylov construction and classification of ergodic Floquet systems
- Krylov Complexity in Lifshitz-type Dirac Field Theories
- Inflationary power spectrum from the Lanczos algorithm
- Krylov Complexity in the Schrödinger Field Theory
- Emergent random matrix universality in quantum operator dynamics
- Complexity Measure Diagnostics of Ergodic to Many-Body Localization Transition
- Krylov Spread Complexity of Quantum-Walks
- Engineering Quantum Reservoirs through Krylov Complexity, Expressivity and Observability
- Krylov complexity of thermal state in early universe
- From Krylov Complexity to Observability: Capturing Phase Space Dimension with Applications in Quantum Reservoir Computing
- Equivalence of dynamics of disordered quantum ensembles and semi-infinite lattices
- Spread Complexity of High Energy Neutrino Propagation over Astrophysical Distances
- Krylov Complexity Under Hamiltonian Deformations and Toda Flows
- Floquet operator dynamics and orthogonal polynomials on the unit circle
- Krylov complexity and Wightman power spectrum with positive chemical potential in Schrödinger field theory
- Krylov Winding and Emergent Coherence in Operator Growth Dynamics
- Higher-Order Krylov State Complexity in Random Matrix Quenches