Quantum Max Cut for complete tripartite graphs
Tea Štrekelj
TL;DR
This work analyzes the quantum Max-d-Cut problem on complete tripartite graphs by exploiting the swap-operator formulation and the algebraic structure of the d-swap algebra via Schur–Weyl duality. By translating the optimization into a representation-theoretic problem, it reduces the d-QMC eigenvalue computation to maximization over partitions constrained by Littlewood–Richardson coefficients, and uses Robin Hood-style box moves to establish height constraints. For d ∈ {2,3}, the paper derives explicit optimal partitions and closed-form expressions for the maximal eigenvalues on $K_{p,q,r}$, revealing precise combinatorial structure governing quantum cut values. The results demonstrate how representation theory can yield exact solutions for specific graph families in quantum many-body problems, offering a tractable approach to otherwise QMA-hard instances in restricted settings.
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, $d \le 3$.