Entanglement Bounds from Cayley Graphs

Cayley graphs offer a useful representation of finitely-generated groups, where vertices represent group elements and edges represent group generators. When considering a group acting on a Hilbert space, the orbit of a quantum state admits a similar graph representation known as a reachability graph. Reachability graphs also encode entanglement information, making them a useful tool for studying state classification and entanglement dynamics in quantum circuits. We introduce a quotient procedure on Cayley graphs which reproduces, and generalizes, state reachability graphs. By abstracting to this operator-level construction, the techniques can be used to study the orbit of any quantum state, and the dynamics of all associated entanglement entropies, under a discrete gate set. We further demonstrate how this quotient protocol can be used to explicitly derive strict bounds on entanglement entropy variation under discrete sets of quantum gates. The generality of our construction enables an analysis of all circuits generated from a finite gate set, including universal gate sets at arbitrary fixed circuit depth.