New eigenvalue bounds unlock better quantum optimization algorithms

Kikuchi graphs are hypergraph constructions whose spectral properties control how well simple product states approximate ground energies of quantum spin systems. By proving the maximum Laplacian eigenvalue of any level-k Kikuchi graph is at most m+k — where m is the edge count of the underlying graph — the authors confirm four recent conjectures and tighten classical approximation algorithms. The results push the approximation ratio for Quantum Max Cut to 0.614 and for the XY Hamiltonian to 0.674, both efficiently achievable. The same technique also advances Brouwer's conjecture on graph Laplacian eigenvalue sums.