In the decade since 2010, successes in artificial intelligence have been at the forefront of computer science and technology, and vector space models have solidified a position at the forefront of artificial intelligence. At the same time, quantum computers have become much more powerful, and announcements of major advances are frequently in the news. The mathematical techniques underlying both these areas have more in common than is sometimes realized. Vector spaces took a position at the axiomatic heart of quantum mechanics in the 1930s, and this adoption was a key motivation for the derivation of logic and probability from the linear geometry of vector spaces. Quantum interactions between particles are modelled using the tensor product, which is also used to express objects and operations in artificial neural networks. This paper describes some of these common mathematical areas, including examples of how they are used in artificial intelligence (AI), particularly in automated reasoning and natural language processing (NLP). Techniques discussed include vector spaces, scalar products, subspaces and implication, orthogonal projection and negation, dual vectors, density matrices, positive operators, and tensor products. Application areas include information retrieval, categorization and implication, modelling word-senses and disambiguation, inference in knowledge bases, and semantic composition. Some of these approaches can potentially be implemented on quantum hardware. Many of the practical steps in this implementation are in early stages, and some are already realized. Explaining some of the common mathematical tools can help researchers in both AI and quantum computing further exploit these overlaps, recognizing and exploring new directions along the way.

中文翻译：

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人工智能中的量子数学

自2010年以来的十年中，人工智能的成功一直处于计算机科学和技术的最前沿，向量空间模型已巩固了在人工智能最前沿的地位。同时，量子计算机变得更加强大，新闻中经常有重大进展的公告。这两个领域的基础数学技术有很多共同点，有时甚至无法实现。向量空间在1930年代是量子力学的公理学领域中的一席之地，这种采用是从向量空间的线性几何推导逻辑和概率的主要动机。使用张量积对粒子之间的量子相互作用进行建模，该张量积还用于表示人工神经网络中的对象和操作。本文介绍了其中一些常见的数学领域，包括在人工智能（AI）中，特别是在自动推理和自然语言处理（NLP）中如何使用它们的示例。讨论的技术包括向量空间，标量积，子空间和蕴涵，正交投影和求反，对偶向量，密度矩阵，正算子和张量积。应用领域包括信息检索，分类和含义，建模词义和歧义消除，知识库中的推理以及语义组成。其中一些方法可能会在量子硬件上实现。此实施中的许多实际步骤尚处于初期阶段，有些已经实现。