# Operator growth bounds from graph theory

## TL;DR

This paper establishes bounds on operator growth in quantum systems using graph theory, improving existing bounds and analyzing scrambling times and chaos in various models, with implications for quantum gravity.

## Contribution

It introduces a topological combinatorial approach to bound operator growth, sharpening Lieb-Robinson bounds and analyzing quantum chaos and scrambling in complex models.

## Key findings

- Bounds on commutator norms via irreducible paths on factor graphs.
- Scrambling time lower bounds of rac{\u221a{ ext{log} N}}{ ext{log} N} in Erdf6s-Re9nyi graphs.
- Upper bounds on Lyapunov exponents close to known results in SYK models.

## 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 $\lVert [A(t),B]\rVert$ 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 $\mathbb{E}\left[ \lVert [A(t),B]\rVert_F^2\right]$. In such quantum systems on Erd\"os-R\'enyi factor graphs, we prove that the scrambling time $t_{\mathrm{s}}$, at which $\lvert [A(t),B]\rVert_F=\mathrm{\Theta}(1)$, is almost surely $t_{\mathrm{s}}=\mathrm{\Omega}(\sqrt{\log N})$; we further prove $t_{\mathrm{s}}=\mathrm{\Omega}(\log N) $ 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.