Sharp Bounds on the Eigenvalues of Kikuchi Graphs and Applications to Quantum Max Cut

TL;DR

This paper establishes upper bounds on the eigenvalues of Kikuchi graph Laplacians, confirming recent conjectures, and applies these results to improve approximation algorithms for Quantum Max Cut and the XY Hamiltonian.

Contribution

It proves new bounds on Kikuchi graph eigenvalues, confirming conjectures, and enhances approximation algorithms for quantum problems.

Findings

Maximum eigenvalue of Kikuchi graph Laplacian is at most m+k.

Tensor products of qubit states achieve 5/8 and 5/7 approximation ratios.

Algorithms achieve 0.614 and 0.674 approximation ratios for Quantum Max Cut and XY Hamiltonian.

Abstract

We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications, we obtain that tensor products of one and two qubit product states achieve an approximation ratio of $5/8$ for Quantum Max Cut and $5/7$ for the XY Hamiltonian. Moreover, combining our bounds with the algorithms analyzed by Apte, Parekh, and Sud, yields efficient algorithms achieving an approximation ratio of $0.614$ for Quantum Max Cut and $0.674$ for the XY Hamiltonian. Finally, we also make modest progress on Brouwer's conjecture and improve Lew's bound on the sum of the top-$k$ eigenvalues of a Graph Laplacian.