Quantum Dynamics in Krylov Space: Methods and Applications

TL;DR

This review explores Krylov subspace methods for analyzing quantum dynamics, emphasizing operator growth, quantum chaos, and applications across many-body systems, quantum field theory, and quantum computing.

Contribution

It provides a comprehensive update on Krylov methods in quantum physics, including recent advances in operator complexity, bounds, and applications to open systems and quantum information.

Findings

Krylov complexity quantifies operator growth and chaos.

Bounds by quantum speed limits relate to operator dynamics.

Applications span quantum control, field theory, and holography.

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.