Quantum Max Cut for complete tripartite graphs

TL;DR

This paper solves the quantum Max-$d$-Cut problem for complete tripartite graphs with small local dimensions, advancing understanding of quantum graph partitioning problems.

Contribution

It provides the first solution to the $d$-QMC problem on complete tripartite graphs for $d \,\leq\, 3$, leveraging algebraic structure insights.

Findings

Solved $d$-QMC for complete tripartite graphs with $d \le 3$

Identified algebraic structures aiding in quantum Hamiltonian analysis

Advances quantum graph partitioning theory

Abstract

The Quantum Max-$d$-Cut ($d$-QMC) problem is a special instance of a $2$-local Hamiltonian problem, representing the quantum analog of the classical Max-$d$-Cut problem. The $d$-QMC problem seeks the largest eigenvalue of a Hamiltonian defined on a graph with $n$ vertices, where edges correspond to swap operators acting on $(\mathbb{C}^d)^{\otimes n}$. In recent years, progress has been made by investigating the algebraic structure of the $d$-QMC Hamiltonian. Building on this approach, this article solves the $d$-QMC problem for complete tripartite graphs for small local dimensions, $d \le 3$.