A Lov\'asz theta lower bound on Quantum Max Cut

TL;DR

This paper establishes a new lower bound for quantum Max Cut using the Lovász theta function, extending classical results and outperforming traditional bounds in quantum scenarios.

Contribution

It introduces a Lovász theta-based lower bound for quantum Max Cut, extending classical bounds and demonstrating its effectiveness for quantum graphs.

Findings

Lower bound for quantum Max Cut in terms of Lovász theta function

Bound achieved by a product state

Outperforms classical bounds on quantum Max Cut

Abstract

We prove a lower bound to quantum Max Cut of a graph in terms of the Lov\'asz theta function of its complement. For a graph with $m$ edges, $\text{qmc}(G) \geq \tfrac{m}{4}\big( 1 + \tfrac{8}{3\pi}\tfrac{1}{\vartheta(\bar{G}) -1} \big)$, with the bound achieved by a product state. The proof extends a result by Balla, Janzer, and Sudakov on classical Max Cut and is also inspired by the randomized rounding method of Gharibian and Parekh. The bound outperforms the classical bound when applied to quantum Max Cut.