Hopf algebras, diagrams and quantum computation

自世纪以来，数学家们就用群论来描述对称性。例如，某些类型的基团捕获晶体的对称性。霍普夫代数理论是群论的推广，它允许描述更复杂系统的对称性，因为它们在同一水平上处理局部和拓扑对称性。除了本身是有趣的数学对象之外，霍普夫代数最近在量子物理和量子计算机科学中有许多应用[2] [3]。科学家经常使用图表来理解或解释他们所研究的系统的行为。在不同的科学领域，经常使用相同类型的图表。在这里，相同的“类型”意味着它们是使用相同的语法绘制的，它们的解释在不同的学科之间是不同的。范畴理论允许形式化这种情况[4]：在某些语法范畴中绘制一个图，而解释是某个语义范畴的函子。在我的博士学位期间，我提议使用范畴理论和图解线性代数来发展Hopf代数理论，以便理解量子物理学，语言学和网络理论的新结构。我的长期目标是使范畴论成为科学界的新语言，使科学领域之间建立联系，并为这些学科创建统一的图像。量子物理学和Hopf代数理论直到最近才用非直观的数学语言表达。量子群研究方法的新奇在于使用了由范畴论和几何学之间的联系所证明（并使之严格）的图解语言。这使得量子物理学成为一个更直观的学科[5]，并将其与计算机科学和语言学等其他科学领域联系起来[6]。该项目与EPSRC的许多研究领域保持一致。首先，Hopf代数的研究对于数学物理领域是重要的，因为它们与二维拓扑量子场论[7]，二维规范理论和量子引力[1]有关。范畴论和图解线性代数是不断增长的兴趣领域，旨在增加科学界的连通性。在量子组，我将与计算机科学家，物理学家和语言学家在跨学科的研究项目。对霍普夫代数的研究也可能在量子技术的发展中取得丰硕成果，因为微软目前正在研究计算的拓扑量子模型[8]，基于表现出霍普夫对称性的量子系统。参考文献[1] Sh.马吉德量子群理论的基础。剑桥大学出版社，1995年。[2]A.基塔耶夫任意子的容错量子计算。Annals Phys.303，第2-30页，2003年。[3]P. Panangeli和E.帕盖特用任意子进行量子计算的分类介绍，物理学讲义第813卷。Springer柏林海德堡，2011年。[4]F：W。劳维尔代数理论的函子语义学。哥伦比亚大学博士论文，1963年。[5]B。Coecke和A.基辛格描绘量子过程。剑桥大学出版社，2017年。[6]B。Coecke，M. Sadrzadeh和S.克拉克意义的组成分布模型的数学基础。eprint arXiv：1003.4394，2010年。[7]B。Balsam和A.基里洛夫Kitaev格点模型与turaev-viro tqfts。eprint arXiv：1206.2308，2012年。[8]伊丽莎白·吉布尼。微软探索拓扑量子计算机的内幕。

Mathematicians have used group theory since the 19th century to describe sym-metry. For instance certain types of groups capture the symmetries of crystals. The theory of Hopf algebras is a generalization of Group theory, which allows describing the symmetries of more complicated systems as they treat local and topological symmetries on the same level. Besides being interesting mathematical objects in their own right [1], Hopf algebras have recently found many applications in quantum physics and quantum computer science [2] [3].Scientists frequently use diagrams to understand or explain the behavior of the systems they study. Very often the same types of diagrams are used in distinct scientific areas. Here the same "type" means that they are drawn using the same syntax, their interpretation then differs between disciplines. Category theory allows formalizing this situation [4]: a diagram is drawn in some syntax category and interpretation is a functor to some semantic category.During my DPhil, I propose to develop the theory of Hopf algebras using category theory and diagrammatic linear algebra in order to understand new structures underlying quantum physics, linguistics and network theory. My long-term objective is to contribute in making category theory the new language in the scientific community to allow connectedness between scientific areas and creating a unified picture of those disciplines.Quantum physics and the theory of Hopf algebras have until recently been expressed in a non-intuitive mathematical language. The novelty of the research methodology in the Quantum group lies in the use of diagrammatic languages justified (and made rigorous) by the connection between category theory and geometry. This has been fruitful in making Quantum physics a more intuitive subject [5] and in relating it to other scientific areas such as computer science and linguistics [6]. This project is in alignment with many of EPSRC's research areas. In first instance the study of Hopf algebras is important for the mathematical physics area as they are known to be related to 2-dimensional topological quantum field theories [7], 2D gauge theories and quantum gravity [1]. Category theory and diagrammatic linear algebra are growing fields of interest aiming to increase the connectedness within the scientific community. In the Quantum group I will work with computer scientists, physicists and linguists in cross-disciplinary research projects. The study of Hopf algebras is also likely to be fruitful in developing quantum technologies as Microsoft is currently working on topological quantum models of computation [8], based on quantum systems that exhibit Hopf symmetries.References[1] Sh. Majid. Foundations of Quantum Group Theory. Cambridge University Press, 1995.[2] A. Kitaev. Fault-tolerant quantum computation by anyons. Annals Phys. 303, pages 2-30, 2003.[3] P. Panangaden and E. Paquette. A categorical presentation of quantum computation with anyons, volume 813 of Lecture Notes in Physics. Springer Berlin Heidelberg, 2011.[4] F: W. Lawvere. Functorial semantics of algebraic theories. PhD thesis, Columbia University, 1963.[5] B. Coecke and A. Kissinger. Picturing Quantum Processes. Cambridge University Press, 2017.[6] B. Coecke, M. Sadrzadeh, and S. Clark. Mathematical foundations for a compositional distributional model of meaning. eprint arXiv:1003.4394, 2010.[7] B. Balsam and A. Kirillov. Kitaev's lattice model and turaev-viro tqfts. eprint arXiv:1206.2308, 2012.[8] Elizabeth Gibney. Inside microsoft's quest for a topological quantum computer.