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 rt(H) = Θ(log N), where routing is via matchings in the clique expansion graph G_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 Θ(log N) routing with L = O(log N) independent overlay layers. An abelian Alon--Boppana barrier shows that fixed-degree Cayley graphs on Z_n² cannot be Ramanujan and affine derandomization on such graphs achieves 15--30% congestion reduction. Towers of k-fold Ramanujan coverings yield (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 ∼4 routing rounds. Displacement energy analyzes greedy adaptive routing, identifying stalling and a hybrid greedy--Valiant protocol achieving ∼3× speedup at practical scales. Hierarchical multi-scale routing achieves O(log² N / log b) depth with boundary-only transfers at capacity k = O(√N log N), and O(log N) depth with optimal block size b = Θ(√n).