Physical, algebraic and geometric underpinnings of topological quantum computation

拓扑量子计算的物理、代数和几何基础

• 批准号: EP/I038683/1
• 负责人: Paul Martin
• 金额: $ 102.37万
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI038683%2F1
• 关键词: Physical; algebraic; geometric; underpinnings; topological

Conventional computer architecture is designed using an essentially classical physical model of the relationship between components and code (hardware and software), in which each `bit' holds a fixed value 0 or 1 until it is changed. Conventional components are large enough that this model is a good approximation to reality. On the other hand, for very small `components' we know that this is _not_ a good approximation. We know from experiment that they behave differently from the classical way (that our experience of the macroscopic world trains us to think about things). This difference can manifest itself as Heisenberg uncertainty, which is not something desirable in a computation. However it can also be thought of, loosely, as taking many values at once, like a hugely fast or massively parallel computer. If this aspect can be harnessed, the disadvantageous `quantum' phenomenon becomes advantageous --- perhaps revolutionarily so. In recent years computer scientists have shown in principle that the parallelism _can_ be harnessed for certain kinds of computation. The next challenge is to design (then build) a quantum computer.However, partly because the quantum model is not intuitive, the best design language is mathematics --- a language built on far fewer `dangerous' assumptions than conventional engineering design. And the good news is (i) that an encouraging basis of usable mathematics is being developed; and (ii) that this challenge is taking the mathematics in intrinsically interesting directions. Leeds University hosts a leading centre for research in quantum information, and also hosts research into some of the main types of mathematics that turn out to be needed: applied representation theory and integrable systems. This project uses expertise in linear category theory, quantum geometry, and related areas of representation theory and integrable systems to provide radically new models of quantum computation. The project interfaces this expertise with expertise on topological phases of matter, and expertise on practitioner constraints, in order to implement the models, ready for laboratory testing. An intriguing way to reinvent the error-robustness of classical digital computing is to work with topological characteristics of the `computer components' --- that is, characteristics that are invariant under small local distortions of the system (which are typically the main kind of error inducing `noise' present). This proposal is concerned, therefore, with the investigation of _topological_ systems that can support quantum information tasks, such as quantum memory, quantum computation and quantum cryptography. The goal is to propose small scale _topological_ models, amenable to laboratory simulations which would then test their feasibility as models for quantum computation. The physics behind the models may be described in terms of `anyon' particles which can be experimentally realized in topological insulators and in graphene carbon, and which can encode and manipulate quantum information error-robustly. The objective here is to develop the theoretical underpinnings of this technology by means of the relation to certain algebraic structures (realized by a topological diagram calculus) and corresponding problems in low-dimensional topology and representation theory. In particular, while guided firmly by the requirements of physical realizability, the project endeavours to deepen the understanding of numerically and analytically solvable models arising from theoretical constructs such as generalized Temperley-Lieb diagram categories, as well as novel models of quantum geometry developed through the theory of exactly integrable quantum systems.

传统的计算机体系结构是使用组件与代码（硬件和软件）之间关系的基本经典物理模型设计的，其中每个“位”都具有固定值0或1，直到更改为止。传统的组件足够大，以至于该模型与现实是一个良好的近似值。另一方面，对于非常小的“组件”，我们知道这是一个很好的近似值。我们从实验中知道，它们的行为与经典的方式不同（我们对宏观世界的经验训练我们思考事物）。这种差异可以表现为海森堡的不确定性，这在计算中不是理想的事情。然而，它也可以轻松地将其视为同时采用许多值，例如一台非常快的或大量平行的计算机。如果可以利用这一方面，那不利的“量子”现象将变得有利 - 也许是革命性的。近年来，计算机科学家原则上表明，对某些计算的平行性_CAN _进行。下一个挑战是设计（然后构建）量子计算机。但是，部分是因为量子模型不是直观的，最佳的设计语言是数学---基于较少的“危险”假设的语言比传统的工程设计要少得多。好消息是（i）正在开发可用数学的令人鼓舞的基础； （ii）这一挑战是将数学朝着本质上有趣的方向发展。利兹大学（Leeds University）主持了量子信息研究的领先研究中心，还主持了对某些数学的主要类型的研究，这些数学是所需的：应用的表示理论和可集成的系统。该项目使用线性类别理论，量子几何形状以及表示理论和可集成系统的相关领域的专业知识来提供量子计算的根本新模型。该项目将这种专业知识与物质拓扑阶段的专业知识和从业者约束的专业知识相结合，以实施模型，准备进行实验室测试。重新发明经典数字计算的误解性的一种有趣的方法是与“计算机组件”的拓扑特征一起使用 - 即在系统的局部小扭曲下不变的特征（通常是诱发“噪声”的主要误差类型）。因此，该建议涉及对可以支持量子信息任务（例如量子内存，量子计算和量子密码学）的_topology_系统的研究。目的是提出小规模_topologicy_模型，可适应实验室模拟，然后将其作为量子计算模型的可行性测试。模型背后的物理学可以用“任何人”粒子来描述，这些粒子可以在拓扑绝缘子和石墨烯碳中实验实现，并且可以编码和操纵量子信息错误。这里的目的是通过与某些代数结构（通过拓扑图计算实现）以及低维拓扑和表示理论的相应问题来开发该技术的理论基础。特别是，虽然根据物理可靠性的要求牢固地指导，但该项目努力加深了对理论构造（如广义templeley-lieb图类别）以及通过准确集成量子系统理论开发的量子几何模型的理论构造，例如广义temperley-lieb图类别，以及新颖的量子几何模型。

项目成果

A generalised Euler-Poincaré formula for associahedra

关联面体的广义欧拉-庞加莱公式

• DOI: 10.48550/arxiv.1711.04986
• 发表时间: 2017
• 期刊: arXiv e-prints
• 作者: Baur Karin

Tonal partition algebras: fundamental and geometrical aspects of representation theory

• DOI: 10.1080/00927872.2023.2239357
• 发表时间: 2019-12
• 期刊: Communications in Algebra
• 影响因子: 0.7
• 作者: C. Ahmed;Paul Martin;V. Mazorchuk

On the number of principal ideals in d-tonal partition monoids

关于 d 调分区幺半群中主理想的数量

• DOI: 10.1007/s00026-020-00518-z
• 发表时间: 2021
• 期刊: Annals of Combinatorics
• 影响因子: 0.5
• 作者: Ahmed C

Winding number order in the Haldane model with interactions

Haldane 模型中相互作用的绕数顺序

• 发表时间: 2015
• 期刊: ArXiv e-prints
• 作者: Alba Emilio

Journeys from quantum optics to quantum technology

• DOI: 10.1016/j.pquantelec.2017.07.002
• 发表时间: 2017-08-01
• 期刊: PROGRESS IN QUANTUM ELECTRONICS
• 影响因子: 11.7
• 作者: Barnett, Stephen M.;Beige, Almut;Kim, M. S.
• 通讯作者: Kim, M. S.