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A<sc>bstract</sc> We study Krylov complexity in various models of quantum field theory: free massive bosons and fermions on flat space and on spheres, holographic models, and lattice models with a UV-cutoff. In certain cases, we observe asymptotic behavior in Lanczos coefficients that extends beyond the previously observed universality. We confirm that, in all cases, the exponential growth of Krylov complexity satisfies the conjectured inequality, which generalizes the Maldacena-Shenker-Stanford bound on chaos. We discuss the temperature dependence of Lanczos coefficients and note that the relationship between the growth of Lanczos coefficients and chaos may only hold for the sufficiently late, truly asymptotic regime, governed by physics at the UV cutoff. Contrary to previous suggestions, we demonstrate scenarios in which Krylov complexity in quantum field theory behaves qualitatively differently from holographic complexity.
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Quantum dynamics in Krylov space: Methods and applications
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, holog- raphy, integrability, quantum control, and quantum computing, as well as current open problems.
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- Award ID(s):
- 2310426
- PAR ID:
- 10608120
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Physics Reports
- Volume:
- 1125-1128
- Issue:
- C
- ISSN:
- 0370-1573
- Page Range / eLocation ID:
- 1 to 82
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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