Operator Growth Bounds from Graph Theory

Abstract

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.

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