Permutation Routing on Ramanujan Hypergraphs with Applications to Neutral Atom Quantum Architectures

TL;DR

This paper studies efficient routing algorithms on Ramanujan hypergraphs with applications to quantum architectures, providing theoretical bounds, novel constructions, and practical protocols for scalable quantum routing.

Contribution

It establishes the routing number of Ramanujan hypergraphs as Θ(log N), introduces new hypergraph constructions, and explores quantum routing protocols with provable efficiency.

Findings

Routing number of Ramanujan hypergraphs is Θ(log N)

Multi-layer stacking achieves Θ(log N) routing with O(log N) layers

Entanglement-assisted routing achieves O(log N) teleportation depth

Abstract

We consider the routing of neutral atoms on a reconfigurable lattice in terms of hypergraph transformations. We prove the routing number of a Ramanujan $(d,r)$-regular hypergraph on $N$ vertices satisfies $\mathrm{rt}(H) = \Theta(\log N)$, where routing is via matchings in the clique expansion graph $G_{\mathrm{cl}}(H)$. Hypergraphs reframe the qubit routing problem by replacing Nenadov's two-sided spectral gap hypothesis with a one-sided condition based on eigenvalue centering. Song--Fan--Miao (SFM) coverings scale for Ramanujan families of every uniformity. A virtual overlay theorem establishes a capacity--depth tradeoff for 3D acousto-optic lens (AOL) architectures, with multi-layer stacking achieving $\Theta(\log N)$ routing with $L = O(\log N)$ independent overlay layers. An abelian Alon--Boppana barrier shows that fixed-degree Cayley graphs on $\mathbb{Z}_n^2$ cannot be Ramanujan and affine derandomization on such graphs achieves 15--30% congestion reduction. Towers of $k$-fold Ramanujan coverings yield $\mathrm(H_L) = O(\log N)$ by recursive routing lift. Entanglement-assisted routing by pre-distributed Bell pairs achieves $O(\log N)$ teleportation depth with a stable crossover at $\sim\!4$ routing rounds. Displacement energy analyzes greedy adaptive routing, identifying stalling and a hybrid greedy--Valiant protocol achieving $\sim\!3\times$ speedup at practical scales. Hierarchical multi-scale routing achieves $O(\log^2 N / \log b)$ depth with boundary-only transfers at capacity $k = O(\sqrt{N} \log N)$, and $O(\log N)$ depth with optimal block size $b = \Theta(\sqrt{n})$.