Spread complexity as classical dilaton solutions

[en] We demonstrate a relation between Nielsen's approach toward circuit complexity and Krylov complexity through a particular construction of quantum state space geometry. We start by associating Kähler structures on the full projective Hilbert space of low rank algebras. This geometric structure of the states in the Hilbert space ensures that every unitary transformation of the associated algebras leave the metric and the symplectic forms invariant. We further associate a classical matter free Jackiw-Teitelboim gravity model with these state manifolds and show that the dilaton can be interpreted as the quantum mechanical expectation values of the symmetry generators. On the other hand, we identify the dilaton with the spread complexity over a Krylov basis thereby proposing a geometric perspective connecting two different notions of complexity.

Research center :

AGIF - Algèbre, Géométrie et Interactions fondamentales

Disciplines :

Physics

Author, co-author :

Chattopadhyay, Arghya ; Université de Mons - UMONS > Faculté des Science > Service de Physique de l'Univers, Champs et Gravitation

Mitra, Arpita ; Department of Physics, Pohang University of Science and Technology, Pohang, South Korea

Van Zyl, Hendrik J. R. ; Laboratory for Quantum Gravity and Strings, Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa ; The National Institute for Theoretical and Computational Sciences, Stellenbosch, South Africa

Language :

English

Title :

Spread complexity as classical dilaton solutions

Publication date :

15 July 2023

Journal title :

Physical Review. D

ISSN :

2470-0010

eISSN :

2470-0029

Publisher :

American Physical Society

Volume :

108

Issue :

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://link.aps.org/article/10.1103/PhysRevD.108.025013

Research unit :

S827 - Physique de l'Univers, Champs et Gravitation

Research institute :

R150 - Institut de Recherche sur les Systèmes Complexes

European Projects :

H2020 - 101034383 - C2W - Connect with Wallonia - Come 2 Wallonia

Funders :

Horizon 2020 Framework Programme
H2020 Marie Skłodowska-Curie Actions
Ministry of Education, Science and Technology
National Research Foundation of Korea
University of the Witwatersrand, Johannesburg
National Research Foundation
National Institute for Theoretical and Computational Sciences
Union Européenne

Funding text :

The work of A. C. is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska Curie Grant Agreement No. 101034383. The work of A. M. is supported by the Ministry of Education, Science, and Technology (No. NRF-2021R1A2C1006453) of the National Research Foundation of Korea (NRF). H. J. R. vZ. is supported by the “Quantum Technologies for Sustainable Development” grant from the National Institute for Theoretical and Computational Sciences (NITHECS). A. C. and H. J. R. vZ. acknowledges the 2021 postdoctoral retreat organized by University of the Witwatersrand with the support of National Research Foundation of South Africa.

Commentary :

31 pages with 2 appendices, v2 has minor restructuring to increase readability matching the published version

Available on ORBi UMONS :

since 29 January 2024

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These operators are denoted as (Equation presented) in Appendix B 1 for the (Equation presented) algebra.
This is special for the low-rank algebras we consider in this article. In general, the Krylov basis will depend on the coefficients appearing in the Hamiltonian, see, e.g., [50].
Though, as we have seen, for this simple group the specific choice of Hamiltonian from the algebra plays almost no role, since the Krylov basis is independent of its coefficients.
This equation matches exactly with (B2) if one maps (Equation presented), where (Equation presented). Therefore we can also see that the dilaton (Equation presented) coincides with the expectation value of the Hamiltonian which is also the isometry generator for the static case.
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In fact, reality condition is sufficient to check that for (Equation presented), η =-2bt-a (-1 +t2 +β2) +c3 (1 +t2 +β2)2β = uk +vkt +wk (t2 +β2)β, under the mapping (Equation presented) with (Equation presented), (Equation presented), (Equation presented), and (Equation presented). Which exactly matches with (B14).
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The general complex solution of the conformal Killing equation in (Equation presented), but for real scalar fields one have to restrict (Equation presented) [60].