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

2026-05-14

AI Summary

This study investigates upper bounds on the largest eigenvalue of the Kikuchi graph Laplacian and their applications in quantum optimization. By integrating techniques from graph theory, spectral graph theory, and quantum information, the authors prove that the largest eigenvalue is at most \(m + k\), thereby verifying four conjectures posed by Apte et al. and improving the known upper bound on the sum of the top \(k\) Laplacian eigenvalues—advancing progress toward resolving the Brouwer conjecture. Leveraging this spectral bound, they design efficient approximation algorithms that achieve theoretical approximation ratios of \(5/8\) for Quantum Max Cut and \(5/7\) for the XY Hamiltonian problem, with empirical performance reaching 0.614 and 0.674, respectively.

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.

Problem

• Kikuchi graphs
• eigenvalue bounds
• Quantum Max Cut
• XY Hamiltonian
• Brouwer's conjecture

Innovation

• Kikuchi graphs
• Laplacian eigenvalues
• Quantum Max Cut
• approximation algorithms
• XY Hamiltonian