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arXiv · 2605.14994

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

Ainesh Bakshi·Arpon Basu·Pravesh Kothari·Anqi Li
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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.

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Ainesh Bakshi, Arpon Basu, Pravesh Kothari, Anqi Li. 2026-05-14. Sharp Bounds on the Eigenvalues of Kikuchi Graphs and Applications to Quantum Max Cut. https://arxiv.org/abs/2605.14994

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