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Integrability and complexity in quantum spin chains
by Ben Craps, Marine De Clerck, Oleg Evnin, Philip Hacker
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|Authors (as registered SciPost users):
|Marine De Clerck
|De Clerck, Marine
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.
Published as SciPost Phys. 16, 041 (2024)
List of changes
We would like to thank both referees for their suggestions and constructive feedback. The manuscript has been updated with the following changes:
- Section 2.1, the introduction to section 3 and section 4.2 have been restructured as requested.
- We corrected the typos pointed out by the second referee, and changed the wording in a few places for clarity (including in the captions as requested).
- We added a reference to  in footnote 1, as well as a new paragraph containing [59-64] at the end of the conclusions section.
In relation to the request by the second referee to clearly contrast what is new compared to , we refer to the subsection "statement of results" in the introduction, which highlights the new results of the present work.
B. Craps, M. De Clerck, O. Evnin, P. Hacker
Submission & Refereeing History
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Same as in first report. With this new version, the readability has improved.
The authors have provided satisfactory replies to my comments, and the presentation has improved, especially in Section 2.1. I am happy to recommend this new version for publication in Scipost Physics.