A relation between Krylov and Nielsen complexity
arXiv:2311.18401 · doi:10.1103/PhysRevLett.132.160402
Abstract
Krylov complexity and Nielsen complexity are successful approaches to quantifying quantum evolution complexity that have been actively pursued without much contact between the two lines of research. The two quantities are motivated by quantum chaos and quantum computation, respectively, while the relevant mathematics is as different as matrix diagonalization algorithms and geodesic flows on curved manifolds. We demonstrate that, despite these differences, there is a relation between the two quantities. Namely, the time average of Krylov complexity of state evolution can be expressed as a trace of a certain matrix, which also controls an upper bound on Nielsen complexity with a specific custom-tailored penalty schedule adapted to the Krylov basis.
v2: published version
References in corpus (16)
- A bound on chaos
- Complexity and Shock Wave Geometries
- Quantum Computation as Geometry
- Circuit complexity in quantum field theory
- Towards Complexity for Quantum Field Theory States
- A Universal Operator Growth Hypothesis
- Quantum chaos and the complexity of spread of states
- Krylov complexity from integrability to chaos
- Krylov Localization and suppression of complexity
- Universal chaotic dynamics from Krylov space
- Quantum Complexity of Time Evolution with Chaotic Hamiltonians
- Krylov Complexity in Free and Interacting Scalar Field Theories with Bounded Power Spectrum
- Complexity Growth in Integrable and Chaotic Models
- Assessing the saturation of Krylov complexity as a measure of chaos
- Bounds on quantum evolution complexity via lattice cryptography
- Integrability and complexity in quantum spin chains
Cited by in corpus (20)
- Quantum Dynamics in Krylov Space: Methods and Applications
- Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos
- Quantum complexity in gravity, quantum field theory, and quantum information science
- Krylov space approach to Singular Value Decomposition in non-Hermitian systems
- Multiseed Krylov complexity
- Krylov complexity of deformed conformal field theories
- Dynamics of monitored SSH Model in Krylov Space: From Complexity to Quantum Fisher Information
- Spread complexity and quantum chaos for periodically driven spin chains
- Measurable Krylov Spaces and Eigenenergy Count in Quantum State Dynamics
- Phase transitions in a non-Hermitian Su-Schrieffer-Heeger model via Krylov spread complexity
- Saturation and recurrence of quantum complexity in random local quantum dynamics
- Properties of Krylov state complexity in qubit dynamics
- Universal Early-Time Growth in Quantum Circuit Complexity
- The Complexity of Being Entangled
- Krylov Spread Complexity of Quantum-Walks
- Engineering Quantum Reservoirs through Krylov Complexity, Expressivity and Observability
- Time-dependent Hamiltonians and Geometry of Operators Generated by Them
- From Krylov Complexity to Observability: Capturing Phase Space Dimension with Applications in Quantum Reservoir Computing
- Spread complexity and localization in -symmetric systems
- Probing the localization effects in Krylov basis