Integrability and complexity in quantum spin chains

Craps, Ben; De Clerck, Marine; Evnin, Oleg; Hacker, Philip

doi:10.21468/SciPostPhys.16.2.041

SciPost Physics

Integrability and complexity in quantum spin chains

Ben Craps, Marine De Clerck, Oleg Evnin, Philip Hacker

SciPost Phys. 16, 041 (2024) · published 7 February 2024

doi: 10.21468/SciPostPhys.16.2.041
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Abstract

There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.16.2.041
TI  - Integrability and complexity in quantum spin chains
PY  - 2024/02/07
UR  - https://scipost.org/SciPostPhys.16.2.041
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 16
IS  - 2
SP  - 041
A1  - Craps, Ben
AU  - De Clerck, Marine
AU  - Evnin, Oleg
AU  - Hacker, Philip
AB  - There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.
ER  -

@Article{10.21468/SciPostPhys.16.2.041,
	title={{Integrability and complexity in quantum spin chains}},
	author={Ben Craps and Marine De Clerck and Oleg Evnin and Philip Hacker},
	journal={SciPost Phys.},
	volume={16},
	pages={041},
	year={2024},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.16.2.041},
	url={https://scipost.org/10.21468/SciPostPhys.16.2.041},
}

Cited by 6

Authors / Affiliations: mappings to Contributors and Organizations

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¹ Ben Craps,
² Marine De Clerck,
¹ ³ Oleg Evnin,
¹ Philip Hacker

Funders for the research work leading to this publication

Fonds Wetenschappelijk Onderzoek (FWO) (through Organization: Fonds voor Wetenschappelijk Onderzoek - Vlaanderen / Research Foundation - Flanders [FWO])
Science and Technology Facilities Council [STFC]
Simons Foundation
Vrije Universiteit Brussel