Quantum dynamics in Krylov space: Methods and applications

Pratik Nandy; Apollonas S. Matsoukas-Roubeas; Pablo Martínez-Azcona; Anatoly Dymarsky; Adolfo del Campo

doi:10.1016/j.physrep.2025.05.001

Quantum dynamics in Krylov space: Methods and applications

Pratik Nandy, Apollonas S. Matsoukas-Roubeas, Pablo Martínez-Azcona, Anatoly Dymarsky, Adolfo del Campo

Research output: Contribution to journal › Review article › peer-review

28 Scopus citations

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.

Original language	English
Pages (from-to)	1-82
Number of pages	82
Journal	Physics Reports
Volume	1125-1128
DOIs	https://doi.org/10.1016/j.physrep.2025.05.001
State	Published - Jun 18 2025

Bibliographical note

Publisher Copyright:
© 2025 Elsevier B.V.

Funding

We are thankful to Budhaditya Bhattacharjee, Hugo A. Camargo, Xiangyu Cao, Paweł Caputa, Nicoletta Carabba, Aurelia Chenu, Pieter W. Claeys, Íñigo L. Egusquiza, Niklas Hörnedal, Norihiro Iizuka, Victor Jahnke, Norman Margolus, Javier Molina-Vilaplana, Masahiro Nozaki, Tanay Pathak, Tomaž Prosen, Zdenek Strakos, Shinsei Ryu, Lucas Sá, Lea F. Santos, Aninda Sinha, Julian Sonner, Ruth Shir, Kazutaka Takahashi, Jing Yang, Zhuo-Yu Xian, and Zhenyu Xu for insightful discussions. We acknowledge financial support from The Luxembourg National Research Fund (project No. grant 17132054 and 16434093, and Attract Grant No. 15382998). One of these projects has received funding from the QUANTERA II Joint Programme with cofunding from the European Union's Horizon Europe research and innovation programme. The work of P.N. is supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Transformative Research Areas (A) “Extreme Universe” No.21H05190. A.D. acknowledges support by the NSF under grant 2310426. For the purpose of open access, the authors have applied a Creative Commons Attribution 4.0 International (CC BY 4.0) license to any Author Accepted Manuscript version arising from this submission. We are thankful to Budhaditya Bhattacharjee, Hugo A. Camargo, Xiangyu Cao, Paweł Caputa, Nicoletta Carabba, Aurelia Chenu, Pieter W. Claeys, Íñigo L. Egusquiza, Niklas Hörnedal, Norihiro Iizuka, Victor Jahnke, Norman Margolus, Javier Molina-Vilaplana, Masahiro Nozaki, Tanay Pathak, Tomaž Prosen, Zdenek Strakos, Shinsei Ryu, Lucas Sá, Lea F. Santos, Aninda Sinha, Julian Sonner, Ruth Shir, Kazutaka Takahashi, Jing Yang, Zhuo-Yu Xian, and Zhenyu Xu for insightful discussions. We acknowledge financial support from The Luxembourg National Research Fund (project No. grant 17132054 and 16434093 , and Attract Grant No. 15382998 ). One of these projects has received funding from the QUANTERA II Joint Programme with cofunding from the European Union’s Horizon Europe research and innovation programme. The work of P.N. is supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Transformative Research Areas (A) “Extreme Universe” No. 21H05190. A.D. acknowledges support by the NSF under grant 2310426. For the purpose of open access, the authors have applied a Creative Commons Attribution 4.0 International (CC BY 4.0) license to any Author Accepted Manuscript version arising from this submission.

Funders	Funder number
European Union's Horizon Europe research and innovation programme
European Union’s Horizon Europe research and innovation programme
National Science Foundation Arctic Social Science Program	CC BY 4.0, 2310426
Fonds National de la Recherche Luxembourg	17132054, 16434093, 15382998
Japan Society for the Promotion of Science	21H05190

Keywords

Krylov complexity
Lanczos algorithm
Operator growth
Quantum chaos

ASJC Scopus subject areas

General Physics and Astronomy

Access to Document

10.1016/j.physrep.2025.05.001

Cite this

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title = "Quantum dynamics in Krylov space: Methods and applications",

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.",

keywords = "Krylov complexity, Lanczos algorithm, Operator growth, Quantum chaos",

author = "Pratik Nandy and Matsoukas-Roubeas, \{Apollonas S.\} and Pablo Mart{\'i}nez-Azcona and Anatoly Dymarsky and \{del Campo\}, Adolfo",

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doi = "10.1016/j.physrep.2025.05.001",

language = "English",

volume = "1125-1128",

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AU - Martínez-Azcona, Pablo

AU - Dymarsky, Anatoly

AU - del Campo, Adolfo

PY - 2025/6/18

Y1 - 2025/6/18

N2 - 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.

AB - 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.

KW - Krylov complexity

KW - Lanczos algorithm

KW - Operator growth

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Quantum dynamics in Krylov space: Methods and applications

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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