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Operator Growth Bounds from Graph Theory

Chen, Chi-Fang and Lucas, Andrew (2021) Operator Growth Bounds from Graph Theory. Communications in Mathematical Physics, 385 (3). pp. 1273-1323. ISSN 0010-3616. doi:10.1007/s00220-021-04151-6.

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Let A and B be local operators in Hamiltonian quantum systems with N degrees of freedom and finite-dimensional Hilbert space. We prove that the commutator norm ∥[A(t),B]∥ is upper bounded by a topological combinatorial problem: counting irreducible weighted paths between two points on the Hamiltonian’s factor graph. Our bounds sharpen existing Lieb–Robinson bounds by removing extraneous growth. In quantum systems drawn from zero-mean random ensembles with few-body interactions, we prove stronger bounds on the ensemble-averaged out-of-time-ordered correlator E[∥[A(t),B]∥²_F]. In such quantum systems on Erdös–Rényi factor graphs, we prove that the scrambling time t_s, at which ∥[A(t),B]∥_F = Θ(1), is almost surely t_s = Ω(√logN); we further prove t_s = Ω(logN) to high order in perturbation theory in 1/N. We constrain infinite temperature quantum chaos in the q-local Sachdev-Ye-Kitaev model at any order in 1/N; at leading order, our upper bound on the Lyapunov exponent is within a factor of 2 of the known result at any q > 2. We also speculate on the implications of our theorems for conjectured holographic descriptions of quantum gravity.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Chen, Chi-Fang0000-0001-5589-7896
Additional Information:© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021. Received 17 May 2019; Accepted 18 June 2021; Published 02 July 2021. This work was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4302, by a Research Fellowship from the Alfred P. Sloan Foundation through Grant FG-2020-13795, and by the Air Force Office of Scientific Research through Grant FA9550-21-1-0195.
Funding AgencyGrant Number
Gordon and Betty Moore FoundationGBMF4302
Alfred P. Sloan FoundationFG-2020-13795
Air Force Office of Scientific Research (AFOSR)FA9550-21-1-0195
Issue or Number:3
Record Number:CaltechAUTHORS:20210803-172928047
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Official Citation:Chen, CF., Lucas, A. Operator Growth Bounds from Graph Theory. Commun. Math. Phys. 385, 1273–1323 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110136
Deposited By: Tony Diaz
Deposited On:04 Aug 2021 18:51
Last Modified:04 Aug 2021 18:51

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