发表机构
Università di Roma “La Sapienza”（罗马大学（罗马第一大学））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出非线性谱计算，利用非线性傅里叶变换在单一物理平台上实现高阶组合优化与布尔逻辑计算，并通过四个输入令牌演示所有256个三变量布尔函数，为模拟计算和机器学习开辟新途径。
AI 中文摘要
许多机器学习算法依赖于傅里叶变换，它也是许多模拟计算设备，特别是光学计算设备的核心。傅里叶谱模的平方自然映射到成对相互作用，如著名的伊辛模型，使其适用于组合优化。在此，我们展示了与可积非线性偏微分方程相关的非线性傅里叶变换如何显著拓宽计算能力，并实现多种计算门。我们特别研究了非线性薛定谔方程的情形；我们引入由“令牌”组成的输入，即编码在多步势中的信息批次，并展示了波传播中产生的非线性谱计算不仅将组合优化推广到更高阶问题，还实现了布尔门。在弱非线性区域，非线性产生高阶相互作用，作为成对项的微扰修正。在孤子区域，离散非线性谱通过孤子生成执行数字计算。通过使用四个输入令牌，我们演示了所有256个三变量布尔函数的可编程性，包括3-SAT实例。我们的结果表明，可积非线性传播为高阶和逻辑计算提供了前所未有的单一物理平台，为模拟计算、机器学习以及经典和量子信息处理开辟了丰富多样的新算法途径。
英文摘要
Many machine learning algorithms rely on the Fourier transform, which is also the inner core of many analog computing devices and, specifically, of optical computing. The square of the modulus of the Fourier spectrum naturally maps onto pairwise interactions, as in the renowned Ising model, making it applicable to combinatorial optimization. Here we show how the nonlinear Fourier transform, related to integrable nonlinear partial differential equations, significantly widens the computational capability and also enables a wide variety of computing gates. We study specifically the case of the nonlinear Schrödinger equation; we introduce an input composed by ``tokens'', i.e., batches of information encoded in a multi-step potential, and we show how the nonlinear spectral computing arising in the wave propagation not only generalizes combinatorial optimization to higher order problems, but also realizes boolean gates. In the weakly nonlinear regime, nonlinearity generates higher-order interactions as perturbative corrections to the pairwise terms. In the solitonic regime, the discrete nonlinear spectrum performs digital computation through soliton generation. By using four input tokens, we demonstrate the programmability of all 256 Boolean functions of three variables, including 3-SAT instances. Our results show that integrable nonlinear propagation provides an unprecedented single physical platform for higher-order and logical computation, opening the way to a rich variety of new algorithms for analog computing, machine learning, and classical and quantum information processing.
