Krylov Subspace Methods for Quantum Dynamics with Time-Dependent Generators

TL;DR

This paper extends Krylov subspace methods to handle quantum systems with time-dependent Hamiltonians, enabling analysis of driven systems and revealing fundamental limits to quantum evolution speed.

Contribution

It introduces a generalized Krylov approach for time-dependent quantum dynamics and connects it to a diffusion problem, establishing new bounds on quantum speed and operator growth.

Findings

Mapped quantum evolution to a diffusion problem in a lattice

Established fundamental limits to quantum speed of evolution

Adapted algorithms for discretized and periodic Hamiltonians

Abstract

Krylov subspace methods in quantum dynamics identify the minimal subspace in which a process unfolds. To date, their use is restricted to time evolutions governed by time-independent generators. We introduce a generalization valid for driven quantum systems governed by a time-dependent Hamiltonian that maps the evolution to a diffusion problem in a one-dimensional lattice with nearest-neighbor hopping probabilities that are inhomogeneous and time dependent. This representation is used to establish a novel class of fundamental limits to the quantum speed of evolution and operator growth. We also discuss generalizations of the algorithm, adapted to discretized time evolutions and periodic Hamiltonians, with applications to many-body systems.