Quantum Dynamics in Krylov Space: Methods and Applications
arXiv:2405.09628 · doi:10.1016/j.physrep.2025.05.001
Abstract
The dynamics of quantum systems unfolds within a subspace of the state space or operator space, known as the Krylov space. This review presents the use of Krylov subspace methods to provide an efficient description of quantum evolution and quantum chaos, with emphasis on nonequilibrium phenomena of many-body systems with a large Hilbert space. It provides a comprehensive update of recent developments, focused on the quantum evolution of operators in the Heisenberg picture as well as pure and mixed states. It further explores the notion of Krylov complexity and associated metrics as tools for quantifying operator growth, their bounds by generalized quantum speed limits, the universal operator growth hypothesis, and its relation to quantum chaos, scrambling, and generalized coherent states. A comparison of several generalizations of the Krylov construction for open quantum systems is presented. A closing discussion addresses the application of Krylov subspace methods in quantum field theory, holography, integrability, quantum control, and quantum computing, as well as current open problems.
Invited review article in Physics Reports, v3: few sections added, typos corrected, refs added. Published version in Physics Reports
References in corpus (294)
- Quantum entanglement
- Supplementary information for "Quantum supremacy using a programmable superconducting processor"
- Holographic Derivation of Entanglement Entropy from AdS/CFT
- Advances in Quantum Metrology
- From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
- A bound on chaos
- Comments on the Sachdev-Ye-Kitaev model
- Adiabatic Quantum Computing
- Non-Markovian dynamics in open quantum systems
- Black holes as mirrors: quantum information in random subsystems
- Symmetry and Topology in Non-Hermitian Physics
- The Magnus expansion and some of its applications
- Building up spacetime with quantum entanglement
- Fast Scramblers
- Shortcuts to adiabaticity: concepts, methods, and applications
- Quantum Non-Markovianity: Characterization, Quantification and Detection
- The distribution of the ratio of consecutive level spacings in random matrix ensembles
- Complexity Equals Action
- Entanglement Renormalization and Holography
- Complexity and Shock Wave Geometries
- Operator Spreading in Random Unitary Circuits
- Fast optimal frictionless atom cooling in harmonic traps
- Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution
- Exact and Approximate Unitary 2-Designs: Constructions and Applications
- Localized shocks
- Complexity, action, and black holes
- Dynamics of Loschmidt echoes and fidelity decay
- Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet
- Phase space representation of quantum dynamics
- Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control
- Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
- A Straightforward Introduction to Continuous Quantum Measurement
- Quantum Computation as Geometry
- Black Holes and Random Matrices
- Thermalization and prethermalization in isolated quantum systems: a theoretical overview
- Biorthogonal Quantum Mechanics
- Traversable Wormholes via a Double Trace Deformation
- Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
- Measuring the scrambling of quantum information
- Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays
- Symmetries and conserved quantities in Lindblad master equations
- Phase-Programmable Gaussian Boson Sampling Using Stimulated Squeezed Light
- Circuit complexity in quantum field theory
- Towards Complexity for Quantum Field Theory States
- Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
- Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator
- Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles
- Strong and weak thermalization of infinite non-integrable quantum systems
- Quantum Zeno dynamics: mathematical and physical aspects
- Evenly distributed unitaries: on the structure of unitary designs
- Diving into traversable wormholes
- Concepts of quantum non-Markovianity: a hierarchy
- Assisted finite-rate adiabatic passage across a quantum critical point: Exact solution for the quantum Ising model
- The Second Law of Quantum Complexity
- Entanglement entropy: holography and renormalization group
- Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
- Minimizing irreversible losses in quantum systems by local counter-diabatic driving
- Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
- Quantum collision models: open system dynamics from repeated interactions
- Chaos, Complexity, and Random Matrices
- Quantum chaos and the complexity of spread of states
- Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories
- Fast atomic transport without vibrational heating
- Does scrambling equal chaos?
- Eternal traversable wormhole
- Floquet-engineering counterdiabatic protocols in quantum many-body systems
- Unifying fast scrambling, thermalization and entanglement through the measurement of FOTOCs in the Dicke model
- Complexity and entanglement for thermofield double states
- Decoherence and the Loschmidt echo
- Operator complexity: a journey to the edge of Krylov space
- Optimal adaptive control for quantum metrology with time-dependent Hamiltonians
- Operator growth in the SYK model
- Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions
- Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos
- On The Evolution Of Operator Complexity Beyond Scrambling
- Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N
- Onset of Random Matrix Behavior in Scrambling Systems
- Les Houches lectures on matrix models and topological strings
- Locality in Quantum and Markov Dynamics on Lattices and Networks
- Towards a full solution of the large N double-scaled SYK model
- An introduction to the SYK model
- Universal Work Fluctuations during Shortcuts To Adiabaticity by Counterdiabatic Driving
- Linear growth of quantum circuit complexity
- Information Scrambling and Loschmidt Echo
- Positive quantum Lyapunov exponents in experimental systems with a regular classical limit
- Digital Quantum Simulation of Minimal AdS/CFT
- Adiabatic tracking of quantum many-body dynamics
- Krylov complexity from integrability to chaos
- Adiabatic eigenstate deformations as a sensitive probe for quantum chaos
- Optimal control, geometry, and quantum computing
- Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK
- Thouless and relaxation time scales in many-body quantum systems
- Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
- Toda chain flow in Krylov space
- Einstein's Equations from Varying Complexity
- Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
- Operator growth and Krylov construction in dissipative open quantum systems
- Krylov Localization and suppression of complexity
- Scrambling the spectral form factor: unitarity constraints and exact results
- Quantum Computational Complexity -- From Quantum Information to Black Holes and Back
- The bulk Hilbert space of double scaled SYK
- Does Complexity Equal Anything?
- Dissipative spin chain as a non-Hermitian Kitaev ladder
- Shortcuts to Adiabaticity in Digitized Adiabatic Quantum Computing
- Universal spectra of random Lindblad operators
- Models of quantum complexity growth
- Many-Body Chaos in the Sachdev-Ye-Kitaev Model
- Quantum chaos: an introduction via chains of interacting spins-1/2
- Universal chaotic dynamics from Krylov space
- Transitionless quantum driving for spin systems
- Digitized-counterdiabatic quantum approximate optimization algorithm
- Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction
- Operator growth in open quantum systems: lessons from the dissipative SYK
- Quantum Krylov subspace algorithms for ground and excited state energy estimation
- Spectral form factors and late time quantum chaos
- Quantum Complexity of Time Evolution with Chaotic Hamiltonians
- Scrambling Dynamics and Out-of-Time Ordered Correlators in Quantum Many-Body Systems: a Tutorial
- Complexity growth of operators in the SYK model and in JT gravity
- Generic dynamical features of quenched interacting quantum systems: Survival probability, density imbalance and out-of-time-ordered correlator
- Phase Transition in the Density of States of Quantum Spin Glasses
- Probing scrambling using statistical correlations between randomized measurements
- Ultimate Speed Limits to the Growth of Operator Complexity
- Symmetry Classification and Universality in Non-Hermitian Many-Body Quantum Chaos by the Sachdev-Ye-Kitaev Model
- Quantum Power Method by a Superposition of Time-Evolved States
- Euclidean operator growth and quantum chaos
- Quantum complexity and topological phases of matter
- The Complexity Geometry of a Single Qubit
- Probing quantum scars and weak ergodicity-breaking through quantum complexity
- Unifying Quantum and Classical Speed Limits on Observables
- Krylov Complexity in Open Quantum Systems
- How to Build the Thermofield Double State
- Holographic Spacetimes as Quantum Circuits of Path-Integrations
- On Krylov complexity in open systems: an approach via bi-Lanczos algorithm
- Krylov complexity and orthogonal polynomials
- Random Matrix Theory for Complexity Growth and Black Hole Interiors
- 2D CFT Partition Functions at Late Times
- A Complexity for Quantum Field Theory States and Application in Thermofield Double States
- Information Scrambling and Chaos in Open Quantum Systems
- Spectral gaps and mid-gap states in random quantum master equations
- Krylov Complexity in Free and Interacting Scalar Field Theories with Bounded Power Spectrum
- Hierarchy of relaxation timescales in local random Liouvillians
- Superadiabatic dynamics in open quantum systems
- Symmetry of Open Quantum Systems: Classification of Dissipative Quantum Chaos
- Renormalized Circuit Complexity
- Shortcuts to Adiabaticity in Driven Open Quantum Systems: Balanced Gain and Loss and Non-Markovian Evolution
- Exact and efficient Lanczos method on a quantum computer
- Krylov complexity and chaos in quantum mechanics
- Thermofield dynamics: Quantum Chaos versus Decoherence
- Aspects of capacity of entanglement
- Towards adiabatic quantum computing using compressed quantum circuits
- Scrambling is Necessary but Not Sufficient for Chaos
- Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2
- Euclidean wormhole in the SYK model
- Lindbladian dissipation of strongly-correlated quantum matter
- Random Lindblad Dynamics
- Resilience of scrambling measurements
- Information scrambling in chaotic systems with dissipation
- Fundamentals of Quantum Mechanics in Liouville Space
- Spread Complexity and Topological Transitions in the Kitaev Chain
- Quantum Gravity in the Lab: Teleportation by Size and Traversable Wormholes
- Many-body quantum teleportation via operator spreading in the traversable wormhole protocol
- Krylov complexity in quantum field theory, and beyond
- Symmetry Classification of Many-Body Lindbladians: Tenfold Way and Beyond
- Spectral Filtering Induced by Non-Hermitian Evolution with Balanced Gain and Loss: Enhancing Quantum Chaos
- Krylov complexity in large- and double-scaled SYK model
- Lifetime of almost strong edge-mode operators in one dimensional, interacting, symmetry protected topological phases
- Dynamics of almost strong edge modes in spin chains away from integrability
- Operator dynamics in Lindbladian SYK: a Krylov complexity perspective
- A statistical mechanism for operator growth
- Spectral and Steady-State Properties of Random Liouvillians
- Portfolio Optimization with Digitized-Counterdiabatic Quantum Algorithms
- Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic Quantum Systems
- Krylov complexity of density matrix operators
- Lindbladian dynamics of the Sachdev-Ye-Kitaev model
- Benchmarking highly entangled states on a 60-atom analog quantum simulator
- Scrambling Transition in a Radiative Random Unitary Circuit
- Iterative Quantum Assisted Eigensolver
- Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field
- Spectral and Krylov Complexity in Billiard Systems
- Exact matrix product solutions in the Heisenberg picture of an open quantum spin chain
- Out-of-time-order correlators in finite open systems
- Universal relation for operator complexity
- Variational principle for optimal quantum controls in quantum metrology
- A universal approach to Krylov State and Operator complexities
- Quantum Gravity in the Lab: Teleportation by Size and Traversable Wormholes, Part II
- Bounds in Nonequilibrium Quantum Dynamics
- Counterdiabatic driving of the quantum Ising model
- Shortcuts to Adiabaticity in Krylov Space
- Digitized-Counterdiabatic Quantum Optimization
- Quantum chaos, scrambling and operator growth in deformed SYK models
- A Holographic Proof of Rényi Entropic Inequalities
- Quantum speed limits on operator flows and correlation functions
- Domain wall dynamics in integrable and chaotic spin-1/2 chains
- Bound on the exponential growth rate of out-of-time-ordered correlators
- Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model
- Pedagogical introduction to SYK model and 2D Dilaton Gravity
- Quantum simulation of quantum field theories as quantum chemistry
- Probing the entanglement of operator growth
- Non-Hermitian Hamiltonian Deformations in Quantum Mechanics
- Long-lived period-doubled edge modes of interacting and disorder-free Floquet spin chains
- Monitoring Quantum Simulators via Quantum Non-Demolition Couplings to Atomic Clock Qubits
- Spread complexity in saddle-dominated scrambling
- Verifying the upper bound on the speed of scrambling with the analogue Hawking radiation of trapped ions
- From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators
- Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos
- State Dependence of Krylov Complexity in CFTs
- Dynamical quantum phase transitions in SYK Lindbladians
- Spread complexity as classical dilaton solutions
- Quantum state complexity meets many-body scars
- On operator growth and emergent Poincaré symmetries
- Weak integrability breaking perturbations of integrable models
- Diagnosing non-Hermitian Many-Body Localization and Quantum Chaos via Singular Value Decomposition
- Geometric Operator Quantum Speed Limit, Wegner Hamiltonian Flow and Operator Growth
- Measuring Spectral Form Factor in Many-Body Chaotic and Localized Phases of Quantum Processors
- A relation between Krylov and Nielsen complexity
- Krylov construction and complexity for driven quantum systems
- Assessing the saturation of Krylov complexity as a measure of chaos
- Decoherence in Conformal Field Theory
- Keldysh Wormholes and Anomalous Relaxation in the Dissipative Sachdev-Ye-Kitaev Model
- Operator growth and Krylov Complexity in Bose-Hubbard Model
- Out-of-time-order fluctuation-dissipation theorem
- On large expansion in the Sachdev-Ye-Kitaev model
- Quantum Information Scrambling: From Holography to Quantum Simulators
- Krylov Complexity in Lifshitz-type Scalar Field Theories
- Singular-Value Statistics of Non-Hermitian Random Matrices and Open Quantum Systems
- Krylov Complexity and Spectral Form Factor for Noisy Random Matrix Models
- Complexity Functionals and Complexity Growth Limits in Continuous MERA Circuits
- Renyi Entropy Dynamics and Lindblad Spectrum for Open Quantum System
- Spread complexity for measurement-induced non-unitary dynamics and Zeno effect
- Dynamical Transition of Operator Size Growth in Quantum Systems Embedded in an Environment
- Measurement master equation
- Signatures of Chaos in Time Series Generated by Many-Spin Systems at High Temperatures
- Quantum bounds and fluctuation-dissipation relations
- Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model
- Weak integrability breaking and level spacing distribution
- Local shortcut to adiabaticity for quantum many-body systems
- Krylov Complexity and Dynamical Phase Transition in the quenched LMG model
- Capacity of Entanglement in Local Operators
- Diffusion constants from the recursion method
- K-GRAPE: A Krylov Subspace approach for the efficient control of quantum many-body dynamics
- Chaos and operator growth in 2d CFT
- Unitarity breaking in self-averaging spectral form factors
- Quantum chaos without false positives
- Refined quantum Lyapunov exponents from replica out-of-time-order correlators
- Operator growth and spread complexity in open quantum systems
- Surprises in the Deep Hilbert Space of all-to-all systems: From super-exponential scrambling to slow entanglement growth
- Proposal for many-body quantum chaos detection
- Tridiagonal Hamiltonians modeling the density of states of the Double-Scaled SYK model
- Inflationary Krylov complexity
- Quantum dynamics in one and two dimensions via recursion method
- Engineered Open Systems and Quantum Simulations with Atoms and Ions
- Krylov complexity for non-local spin chains
- The operator growth hypothesis in open quantum systems
- Thermalization in Krylov Basis
- Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems
- The growth of operator entropy in operator growth
- Probing quantum chaos in multipartite systems
- Krylov Complexity of Open Quantum Systems: From Hard Spheres to Black Holes
- Exact universal bounds on quantum dynamics and fast scrambling
- Krylov complexity of deformed conformal field theories
- Quantum delocalization on correlation landscape: The key to exponentially fast multipartite entanglement generation
- Universal Hypothesis of Autocorrelation Function from Krylov Complexity
- Q-curvature and Path Integral Complexity
- Krylov Subspace Methods for Quantum Dynamics with Time-Dependent Generators
- Low temperature quantum bounds on simple models
- Stochastic Operator Variance: an observable to diagnose noise and scrambling
- A Universal Model of Floquet Operator Krylov Space
- Estimation of equilibration time scales from nested fraction approximations
- Measurable Krylov Spaces and Eigenenergy Count in Quantum State Dynamics
- Universal Properties of the Spectral Form Factor in Open Quantum Systems
- Variational adiabatic transport of tensor networks
- Quantum Velocity Limits for Multiple Observables: Conservation Laws, Correlations, and Macroscopic Systems
- Black Holes and Marchenko-Pastur Distribution
- Analyticity constraints bound the decay of the spectral form factor
- Operator growth in SU(2) Yang-Mills theory
- Operator size growth in Lindbladian SYK
- Quantifying operator spreading and chaos in Krylov subspaces with quantum state reconstruction
- A Lanczos approach to the Adiabatic Gauge Potential
- Defining classical and quantum chaos through adiabatic transformations
- Operator growth in a quantum compass model on a Bethe lattice
- Quantum chaos, integrability, and late times in the Krylov basis
- Krylov Spread Complexity of Quantum-Walks
- Spread complexity and the saturation of wormhole size
- Complexity enriched dynamical phases for fermions on graphs
- Generalised Krylov complexity
- Weak vs. strong breaking of integrability in interacting scalar quantum field theories
- Dissecting Quantum Many-body Chaos in the Krylov Space
- Proof of a Universal Speed Limit on Fast Scrambling in Quantum Systems
- Inflationary thermal Krylov complexity
- Topological Origin of Floquet Thermalization in Periodically Driven Many-body Systems
- Lanczos spectrum for random operator growth
- Scaling Relations of Spectrum Form Factor and Krylov Complexity at Finite Temperature
- Towards complexity in de Sitter space from the double-scaled Sachdev-Ye-Kitaev model
- Operator Krylov complexity in random matrix theory
Cited by in corpus (54)
- Krylov Complexity in Mixed Phase Space
- Tuning the Quantum Mpemba Effect in Isolated System by Initial State Engineering
- Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field
- Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information
- Pseudomode expansion of many-body correlation functions
- Phase transitions in a non-Hermitian Su-Schrieffer-Heeger model via Krylov spread complexity
- Exactly solvable models for universal operator growth
- Diagnosing Quantum Many-body Chaos in Non-Hermitian Quantum Spin Chain via Krylov Complexity
- Toward Krylov-based holography in double-scaled SYK
- Almost Strong Zero Modes at Finite Temperature
- Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics
- Krylov operator complexity in holographic CFTs: Smeared boundary reconstruction and the dual proper radial momentum
- Complexity and Operator Growth in Holographic 6d SCFTs
- Approximation theory for Green's functions via the Lanczos algorithm
- Free Probability approach to spectral and operator statistics in Rosenzweig-Porter random matrix ensembles
- Krylov Complexity in Lifshitz-type Dirac Field Theories
- Inflationary power spectrum from the Lanczos algorithm
- Lanczos-Pascal approach to correlation functions in chaotic quantum systems
- Holographic Krylov complexity in the Coulomb branch of SYM
- Multi-Qubit Parity Gates for Rydberg Atoms in Various Configurations
- Emergent random matrix universality in quantum operator dynamics
- Dynamical quantum phase transition, metastable state, and dimensionality reduction: Krylov analysis of fully connected spin models
- Exact Model Reduction for Continuous-Time Open Quantum Dynamics
- Quantum Chaos, Randomness and Universal Scaling of Entanglement in Various Krylov Spaces
- Escaping the Krylov space during finite precision Lanczos
- Entropy and multiplicity of hadrons in the high energy limit within dipole cascade models
- Multifractality in high-dimensional graphs induced by correlated radial disorder
- Universal Time Evolution of Holographic and Quantum Complexity
- A time-marching quantum algorithm for simulation of the nonlinear Lorenz dynamics
- Absence of nontrivial local conserved quantities in the quantum compass model on the square lattice
- Engineering Quantum Reservoirs through Krylov Complexity, Expressivity and Observability
- Krylov complexity of purification
- Signatures of quantum chaos and complexity in the Ising model on random graphs
- On the stabilizer complexity of Hawking radiation
- Complexity of Quadratic Quantum Chaos
- Connecting Magic Dynamics in Thermofield Double States to Spectral Form Factors
- Transport approach to quantum state tomography
- Classical periodic trajectories and quantum scars in many-spin systems
- Krylov-space anatomy and spread complexity of a disordered quantum spin chain
- On the reconstruction map in JT gravity
- Dynamics of Loschmidt echoes from operator growth in noisy quantum many-body systems
- From Krylov Complexity to Observability: Capturing Phase Space Dimension with Applications in Quantum Reservoir Computing
- Floquet operator dynamics and orthogonal polynomials on the unit circle
- Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism
- Topological crystals and soliton lattices in a Gross-Neveu model with Hilbert-space fragmentation
- Phenomenological model of decaying Bose polarons
- Krylov complexity and Wightman power spectrum with positive chemical potential in Schrödinger field theory
- A graph-theoretic approach to chaos and complexity in quantum systems
- Krylov Winding and Emergent Coherence in Operator Growth Dynamics
- Krylov Complexity Under Hamiltonian Deformations and Toda Flows
- Higher-Order Krylov State Complexity in Random Matrix Quenches
- Resonant level model from a Krylov perspective: Lanczos coefficients in a quadratic model
- Information acquisition, scrambling, and sensitivity to errors in quantum chaos
- Long-Range Pairing in the Kitaev Model: Krylov Subspace Signatures