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和代码（硬件和软件）之间关系的基本上经典的物理模型设计的，其中每个“位”保持固定值0或1，直到它被改变。传统的组件足够大，这个模型是一个很好的近似现实。另一方面，对于非常小的"组件"，我们知道这不是一个很好的近似。我们从实验中知道，它们的行为与经典方式不同（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不确定性，这在计算中是不可取的。然而，它也可以被认为是，松散地，作为一次采取许多值，就像一个非常快或大规模并行计算机。如果能利用这一点，不利的“量子”现象就会变得有利--也许是革命性的。近年来，计算机科学家已经在原则上证明了并行性可以用于某些类型的计算。下一个挑战是设计（然后建造）一台量子计算机，然而，部分原因是量子模型不是直观的，最好的设计语言是数学-一种建立在比传统工程设计少得多的"危险"假设上的语言。好消息是（i）一个令人鼓舞的可用数学基础正在发展;（ii）这个挑战正在把数学带到本质上有趣的方向。利兹大学拥有一个领先的量子信息研究中心，并主持了对一些主要数学类型的研究，这些数学类型被证明是必要的：应用表示论和可积系统。该项目使用线性范畴理论，量子几何，表示论和可积系统的相关领域的专业知识，提供量子计算的全新模型。该项目将这一专业知识与物质拓扑阶段的专业知识和从业人员限制的专业知识相结合，以便实施模型，为实验室测试做好准备。一个有趣的方法来重塑经典的数字计算的错误鲁棒性是工作与拓扑特征的“计算机组件”-也就是说，特征是不变的小局部失真的系统（这是典型的主要类型的错误诱导“噪音”目前）。因此，该建议关注的是研究能够支持量子信息任务的拓扑系统，如量子存储，量子计算和量子密码学。我们的目标是提出小尺度拓扑模型，服从实验室模拟，然后将测试其作为量子计算模型的可行性。模型背后的物理学可以用“任意子”粒子来描述，这些粒子可以在拓扑绝缘体和石墨烯碳中实验实现，并且可以编码和操纵量子信息。这里的目标是通过与某些代数结构（由拓扑图演算实现）的关系以及低维拓扑和表示论中的相应问题来发展这种技术的理论基础。特别是，在严格遵循物理可实现性要求的同时，该项目致力于加深对从理论结构（如广义Temperley-Lieb图类别）中产生的数值和解析可解模型的理解，以及通过精确可积量子系统理论开发的量子几何新模型。

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

Closed-form modified Hamiltonians for integrable numerical integration schemes

• DOI: 10.1088/1361-6544/aad9ac
• 发表时间: 2017-07
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Shami A. M. Alsallami;Jitse Niesen;F. Nijhoff

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