在拉马努金超图上的排列路由及其在中性原子量子架构中的应用
Permutation Routing on Ramanujan Hypergraphs with Applications to Neutral Atom Quantum Architectures
AI总结：
本文研究了中性原子在可重构晶格中的路由问题，证明了拉马努金（d,r）-正则超图的路由数为Θ(log N)，并探讨了其在中性原子量子架构中的应用。
摘要：
我们考虑了中性原子在可重构晶格中的路由问题，通过超图变换来分析。我们证明了具有N个顶点的拉马努金(d,r)-正则超图的路由数满足rt(H)=Θ(log N)，其中路由是通过在 clique 扩张图G_cl(H)中的匹配进行的。超图将qubit路由问题重新表述为基于特征值中心的单侧条件，取代了Nenadov的双侧谱间隙假设。Song-Fan-Miao (SFM) 覆盖对于拉马努金家族的每个均匀性都可扩展。一个虚拟叠加定理建立了3D声光透镜(AOL)架构的容量-深度权衡，多层堆叠实现Θ(log N)的路由，使用L=O(log N)个独立叠加层。一个阿贝尔Alon-Boppana障碍表明，固定度数的Cayley图在Z_n^2上不能是拉马努金的，且在这些图上的仿射去随机化可实现15-30%的拥堵减少。k重拉马努金覆盖的塔产生H_L=O(log N)通过递归路由提升。通过预分配的贝尔对进行的纠缠辅助路由实现O(log N)的量子态传送深度，具有稳定的交叉点在约4轮路由中。位移能量分析了贪心自适应路由，识别了停滞现象，并发现混合贪心-Valiant协议在实际规模上实现约3倍的加速。分层多尺度路由在容量k=O(√N log N)时实现O(log²N/log b)的深度，仅通过边界转移，并在最优块大小b=Θ(√n)时实现O(log N)的深度。
英文摘要：
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 $\mathrm{rt}(H) = Θ(\log N)$, 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 $Θ(\log N)$ 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 $b = Θ(\sqrt{n})$.
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