Category Theory, Artificial Intelligence and Interdisciplinary Quantum Structures

范畴论最初是纯数学的一个分支，但由于其实际应用而日益受到人们的关注。应用领域广泛，包括但不限于：编程语言设计，数据库管理，基础物理，量子计算，人工智能，语言学及其交叉。范畴论是一种富有表现力而又严谨的语言，能够孤立地描述这些现象中的每一种，但也是通过设计来绘制它们之间联系的理想语言。这要归功于它在历史上用于将看似不同的数学分支联系起来，此后它已发展成为物理学和计算机科学等领域的跨学科应用。该项目可能影响的一个领域是量子计算。最近的研究基于范畴理论的图形语言已被用来大大减少量子电路的大小。在这个方向上的进一步研究可能有助于生产更高效的量子计算机，加速量子计算的大规模发展。另一个潜在的影响是人工智能。目前，主流的人工智能主要由“黑盒”实现组成：决策过程通常是基于知识的，通常缺乏理解和可解释性。范畴理论适合于形式化抽象过程，可以更好地理解人工智能。此外，在人工智能和量子计算的交叉点上，最近的发展表明，通过用范畴理论描述语言和量子电路，一些自然语言处理任务很容易转换为量子电路。这预示着量子计算机在未来更广泛的人工智能任务中的应用。目标和目的包括：探索图形语言在量子计算中的应用，如量子电路优化，量子软件和量子算法。设计经典和量子人工智能算法的表达性，但形式化的范畴论描述，使它们更适合人类理解。设计基于范畴论的方法，在人工智能应用中利用量子计算。进一步发展数学形式主义，揭示数学，计算、基础物理学和认知。这项研究的新颖之处：目前关于量子电路图形语言的许多工作都使用了一种称为ZX演算的特定语言。尽管如此，还存在其他变体，例如“ZW”和“ZH”。不同的语言可能更适合不同的应用程序。因此，该项目的范围不仅限于ZX，还寻求探索其他图形语言的新用途和发展。许多人工智能研究都以结果为中心，但缺乏对产生这些结果的智能系统的可解释性和理解。该项目采取了另一种方法，优先考虑更深入地了解这些系统为什么工作，为什么它们表现得那么好，并在它们所做的决策和预测背后寻求人类可解释的理由。量子人工智能仍然局限于机器学习和自然语言处理应用的狭窄范围。另一方面，在量子领域，人工智能还有大量的子领域有待探索，例如计算机视觉和强化学习。该项目旨在将量子人工智能任务的当前范围扩展到超出其当前限制的范围，发现新的用例。该项目是跨学科的，涉及EPSRC数学科学研究主题，人工智能技术研究领域和量子技术主题的各个方面。

Originally a branch of Pure Mathematics, Category Theory is attracting increasing attention for practical uses. Application areas are broad, including, but not limited to: Programming Language Design, Database Management, Fundamental Physics, Quantum Computing, Artificial Intelligence, Linguistics and intersections thereof. Category Theory is an expressive yet rigorous language, able to describe each of these phenomena in isolation, but also ideal for drawing connections between them by design. This owes to its historical use for relating seemingly disparate branches of Mathematics, which has since evolved into more interdisciplinary uses across fields such as Physics and Computer Science.An area this project potentially impacts is Quantum Computing. Recent research on Category Theory-based diagrammatic languages has been used to drastically reduce the size of Quantum Circuits. Further research in this direction may help produce more efficient Quantum Computers, accelerating the development of Quantum Computation at scale.Another potential impact is on Artificial Intelligence. Currently, mainstream Artificial Intelligence largely consists of 'black box' implementations: decision making procedures are frequently heuristics-based and often lack understanding and interpretability. Suitable for formalising abstract processes, Category Theory may allow for better understood Artificial Intelligence.Furthermore, at the intersection of Artificial Intelligence and Quantum Computing, recent developments have shown that by describing both language and Quantum circuits in terms of Category Theory, some Natural Language Processing tasks easily translate to Quantum circuits. This bodes well for future applications of Quantum Computers to a wider variety of Artificial Intelligence tasks.Aims and objectives include:Explore the applications of diagrammatic languages to Quantum Computation such as Quantum circuit optimization, Quantum software and Quantum algorithms.Devise expressive, yet formal Category-theoretic descriptions of both Classical and Quantum Artificial Intelligence algorithms that make them more amenable to human understanding.Devise Category Theory based methods for utilizing Quantum Computation in Artificial Intelligence applications.Further develop Mathematical formalisms to uncover connections between Mathematics, Computation, Fundamental Physics and Cognition.Ways in which this research is novel:Much of the current work concerning diagrammatic languages for Quantum circuits utilises a specific language called the ZX-Calculus. Despite this, other variations exist, such as 'ZW' and 'ZH'. Different languages may be more suitable for different applications. As such, this project is not limited in scope to ZX, but also seeks to explore novel use and development of other diagrammatic languages.Much Artificial Intelligence research is results-focused but lacks interpretability and understanding of the intelligent systems producing these results. This project takes an alternative approach, prioritizing a deeper understanding of why these systems work, why they perform as well as they do, and to seek a human-interpretable rationale behind the decisions and predictions they make.Quantum Artificial Intelligence is still limited to a narrow range of Machine Learning and Natural Language Processing applications. On the other hand, there is a wealth of sub-fields of Artificial Intelligence that have yet to be explored in the Quantum realm, such as Computer Vision and Reinforcement Learning. This project aims to extend the current scope of Quantum Artificial Intelligence tasks beyond its current limits, uncovering novel use cases.This project is interdisciplinary, addressing aspects of the EPSRC Mathematical Sciences research theme, the Artificial Intelligence Technologies research area, and the Quantum Technologies theme.