Operator growth bounds from graph theory

Understand

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 $\mathbb{E}\left[ \lVert [A(t),B]\rVert_F^2\right]$.