有向逻辑中嵌套级联的理论

中文摘要 AI 辅助

在有向逻辑（DL）中，电子控制的光学元件充当光子门。这类电光元件具有混合性质：它们有两个输入——电子输入和光学输入——但只有一个光学输出。这使得在不进行光学与电子表示之间反复转换的情况下级联此类门变得繁琐。Shamir和Hardy提出的嵌套级联方案[Opt. Express, 17, 150 (2007)]在很大程度上可以克服这一问题。尽管该方案有望解决级联问题，但迄今为止仅对少数最简单的情况进行了证明或实现。在此，我们发展了一个适用于任意数量门的嵌套级联的一般严格理论。我们提出了一种易于扩展到大量门的算法变体，并严格证明了其有效性。此外，我们分析了嵌套级联如何随相应布尔公式的规模扩展。我们表明，在许多重要情况下，良好的（线性）可扩展性得到保证，而相对于相应布尔公式的平均扩展仅为适度多项式，指数约为3/2。然而，对于允许共享和重用中间结果的更一般的布尔电路，最坏情况下的扩展仍然是指数级的。

英文摘要

In directed logic (DL), electronically controlled optical elements serve as photonic gates. Such electro-optical elements are of mixed nature: they have two inputs--electronic and optical--but only one, optical, output. This makes cascading such gates without repeated conversion between optical and electronic representations cumbersome. This problem can be largely overcome using the nested cascading scheme proposed by Shamir and Hardy [Opt. Express, 17, 150 (2007)]. Although promising as a solution to the cascading problem, the Shamir-Hardy scheme has so far been proved or implemented only for a few simplest cases. Here, we develop a general rigorous theory of nested cascading, valid for an arbitrary number of gates. We propose a variant of the algorithm which easily extendable to large number of gates, and rigorously prove its validity. Furthermore, we analyze how nested cascading scales with the size of the corresponding Boolean formula. We show that good (linear) scalability is guaranteed in many important cases, while the average scaling with respect to the corresponding Boolean formulas is only moderately polynomial, with an exponent of approximately 3/2. Yet, the worst-case scaling remains exponential with respect to more general Boolean circuits allowing sharing and reuse of intermediate results.

发表机构

• Institute of Quantum Optics, Leibniz University Hannover（汉诺威莱布尼茨大学量子光学研究所）
• Max Born Insitute, Berlin（柏林马克斯·玻恩研究所）
• Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering - Innovation Across Disciplines), Leibniz University Hannover（汉诺威莱布尼茨大学凤凰D卓越集群（光子学、光学与工程——跨学科创新））

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