Hopf algebras, diagrams and quantum computation

自19世纪以来，数学家一直使用群论来描述对称性。例如，某些类型的基团捕捉晶体的对称性。Hopf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对称性和拓扑对称性。除了本身是有趣的数学对象之外，Hopf代数最近在量子物理和量子计算机科学中发现了许多应用。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，因此它们的解释在学科之间是不同的。范畴论允许形式化这种情况[4]：在某个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，我建议使用范畴论和图解线性代数来发展Hopf代数理论，以理解量子物理学、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，从而使科学领域之间建立联系，并为这些学科创造一个统一的图景。直到最近，量子物理学和霍普夫代数理论都是用一种非直观的数学语言来表达的。量子小组研究方法的新颖之处在于使用图解语言，通过范畴论和几何之间的联系来证明（并使之严格）。这使得量子物理学成为一门更直观的学科，并将其与其他科学领域（如计算机科学和语言学）联系起来。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很重要，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人们越来越感兴趣的领域，旨在增加科学界的连通性。在量子小组中，我将与计算机科学家、物理学家和语言学家一起进行跨学科的研究项目。Hopf代数的研究也可能在开发量子技术方面取得丰硕成果，因为微软目前正在基于表现出Hopf对称性的量子系统研究计算的拓扑量子模型[8]。参考文献[1]Sh. Majid。量子群论基础。剑桥大学出版社，1995答:Kitaev。任意子的容错量子计算。《物理学报》2003年第3期，第2-30页P. Panangaden和E. Paquette。用任意子进行量子计算的分类描述，《物理学讲义》第813卷。施普林格柏林，海德堡，2011F: W. Lawvere。代数理论的功能语义。博士论文，哥伦比亚大学，1963年B.科克和A.基辛格。描绘量子过程。剑桥大学出版社，2017B. Coecke， M. Sadrzadeh， S. Clark。意义的组成分布模型的数学基础。[j] .电子信息学报，2010,31 (4):391 - 391B.巴尔萨姆和A.基里洛夫。Kitaev的晶格模型和turaev-viro模型。[j] .电子科学学报，2012,32 (2):481 - 481伊丽莎白·吉。微软对拓扑量子计算机的探索。

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.