A Lovász theta lower bound on Quantum Max Cut

TL;DR

This work gives a quantum analogue of a classical Max Cut bound by expressing a lower bound on Quantum Max Cut (QMC) in terms of the Lovász theta function of a graph's complement. The authors leverage a Gaussian random projection rounding framework (Briët–de Oliveira Filho–Vallentin) and a product-state construction to translate theta-realizing vectors into quantum states, yielding a bound $qmc(G) \ge \frac{m}{4}\left(1 + \frac{8}{3\pi}\frac{1}{\vartheta(\overline{G})-1}\right)$. The bound is achieved by a product state and improves upon the classical bound when applied to QMC, with extensions to XX-type interactions and connections to the vector chromatic number. The work integrates hypergeometric function analysis and rounding techniques to bridge classical and quantum approximations for graph partitioning problems.

Abstract

We prove a lower bound to quantum Max Cut of a graph in terms of the Lovász theta function of its complement. For a graph with $m$ edges, $\text{qmc}(G) \geq \tfrac{m}{4}\big( 1 + \tfrac{8}{3π}\tfrac{1}{\vartheta(\bar{G}) -1} \big)$, 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.

Theorems & Definitions (7)

• Definition 1: 10.1145/274787.274791
• Lemma 2: v010a004, Lemma 2.1
• Lemma 3
• proof
• Theorem 4
• proof
• Definition 5: 10.1145/274787.274791