Hopf algebras, diagrams and quantum computation

自19世纪以来，数学家们一直使用群论来描述对称。例如，某些类型的基团捕捉到了晶体的对称性。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，就像它们在同一水平上处理局部对称性和拓扑对称性一样。除了本身就是有趣的数学对象外，Hopf代数最近在量子物理和量子计算机科学中得到了许多应用[2][3]。科学家们经常使用图表来理解或解释他们研究的系统的行为。在不同的科学领域中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]：在某个语法范畴中画出一个图，而解释是某个语义范畴的函数器。在我的DPhil中，我建议使用范畴理论和图形线性代数来发展Hopf代数理论，以便理解量子物理、语言学和网络理论背后的新结构。我的长期目标是使范畴理论成为科学界的一种新语言，允许科学领域之间的联系，并创建这些学科的统一图景。直到最近，量子物理学和Hopf代数理论都是用一种非直观的数学语言表达的。量子学派的研究方法的新奇之处在于使用了图解语言，范式论和几何之间的联系证明了这一点(并使其变得严谨)。这使得量子物理学成为一门更直观的学科[5]，并将其与其他科学领域，如计算机科学和语言学[6]联系起来，取得了丰硕成果。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。范畴理论和图形线性代数是人们日益感兴趣的领域，旨在增加科学界的联系。在量子小组中，我将与计算机科学家、物理学家和语言学家在跨学科研究项目中合作。霍普夫代数的研究也可能在发展量子技术方面取得丰硕成果，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的量子系统。马吉德。量子群论的基础。剑桥大学出版社，1995.[2]A.Kitaev。任意子容错量子计算。物理年鉴。303，2-30页，2003年。任意子量子计算的分类介绍，物理讲义第813卷。施普林格·柏林·海德堡，2011。代数理论的函数语义学。哥伦比亚大学博士论文，1963年。[5]B.科克和A.基辛格。想象量子过程。剑桥大学出版社，2017。[6]B.Coecke，M.Sadrzadeh，S.Clark。意义成分分布模型的数学基础。EPrint arxiv：1003.4394,2010年.[7]B.Balsam和A.Kitaev的格子模型和Turaev-Viro模型。EPrint arxiv：1206.2308,2012年。在微软寻求拓扑量子计算机的过程中。

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.