Analogy between Boltzmann Machines and Feynman Path Integrals
Analogy between Boltzmann Machines and Feynman Path Integrals
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DOI：
10.1021/acs.jctc.3c00187
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发表时间：
2023-04-26
影响因子：
5.5
通讯作者：
Kais, Sabre
中科院分区：
文献类型：
--
作者：
Iyengar, Srinivasan S.;Kais, Sabre
机器学习对科学、技术、健康以及计算机和信息科学的多个领域产生了重大影响。随着量子计算的出现，量子机器学习已经发展成为研究复杂学习问题的一种新的重要途径。然而，关于机器学习的基础，人们存在着大量的争论和不确定性。在这里，我们提供了一种称为玻尔兹曼机器的一般机器学习方法与费曼对量子力学和统计力学的描述之间的数学联系的详细阐述。在费曼的描述中，量子现象产生于路径上优雅的加权和(或叠加)。我们的分析表明，Boltzmann机器和神经网络具有相似的数学结构。这允许将Boltzmann机器和神经网络中的隐藏层解释为路径元素的离散版本，并允许对机器学习的路径积分解释类似于量子力学和统计力学中的解释。由于费曼路径是对干涉现象和与量子力学密切相关的叠加原理的自然而优雅的描述，这种分析允许我们将机器学习的目标解释为通过网络找到路径和累积的路径权重的适当组合，从而累积地捕获给定数学问题的x-to-y映射的正确属性。我们被迫得出结论，神经网络与费曼路径积分自然相关，因此可能提供了一种被认为是量子问题的途径。因此，我们提供了适用于Boltzmann机器和Feynman路径积分的一般量子电路模型。
Machine learning has had a significant impact on multiple areas of science, technology, health, and computer and information sciences. Through the advent of quantum computing, quantum machine learning has developed as a new and important avenue for the study of complex learning problems. Yet there is substantial debate and uncertainty in regard to the foundations of machine learning. Here, we provide a detailed exposition of the mathematical connections between a general machine learning approach called Boltzmann machines and Feynman's description of quantum and statistical mechanics. In Feynman's description, quantum phenomena arise from an elegant, weighted sum over (or superposition of) paths. Our analysis shows that Boltzmann machines and neural networks have a similar mathematical structure. This allows the interpretation that the hidden layers in Boltzmann machines and neural networks are discrete versions of path elements and allows a path integral interpretation of machine learning similar to that in quantum and statistical mechanics. Since Feynman paths are a natural and elegant depiction of interference phenomena and the superposition principle germane to quantum mechanics, this analysis allows us to interpret the goal in machine learning as finding an appropriate combination of paths, and accumulated path-weights, through a network, that cumulatively captures the correct properties of an x-to -y map for a given mathematical problem. We are forced to conclude that neural networks are naturally related to Feynman path-integrals and hence may present one avenue to be considered as quantum problems. Consequently, we provide general quantum circuit models applicable to both Boltzmann machines and Feynman path integrals.