Collaborative Research: Topological Phases of Matter and their Application to Quantum Computing

合作研究：物质的拓扑相及其在量子计算中的应用

• 批准号: 1108725
• 负责人: Eric Rowell
• 金额: $ 19.33万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108725&HistoricalAwards=false
• 关键词: Collaborative; Research; Topological; Phases; Matter

In this collaborative project the Principal Investigators (PIs) willinvestigate mathematical models for topological phases of matter and theirapplication to the nascent field of quantum information. Examples oftopological phases of matter include Fractional Quantum Hall liquids andtopological insulators: materials subjected to extreme physical conditions(e.g. near zero Kelvin, under large magnetic forces) which appear toexhibit topological behavior. The main objectives of the project are tobetter understand the taxonomy of these exotic states of matter throughmathematical models, address physically and computationally motivatedmathematical problems, and apply topological and algebraic methods tostudy them. For example, the issue of quantum computational power(universality) may be analyzed by finding the images of the braid groupsunder representations associated to a particular model. Moreover, theextant hypothesized models for topological states of matter do not captureall of the subtleties (such as fermionic topological order) so the PIswill develop new models.The classical states of matter of solid, liquid and gas have more refinedclassifications: for example, solid crystals may be differentiated bytheir symmetries. Newly discovered topological materials have yet to befully understood, but potentially can be used to build (quantum)computational devices out-performing standard micro-chip based computers.The most commonly encountered model for quantum computation, the quantumcircuit model, requires challenging, if not impossible, accuracy on thehardware to be of practical value due to local interactions of the systemwith the surrounding environment. The topological model based on exoticstates of matter, while mathematically more complicated, has a built-intolerance for such interactions. The PIs will mathematically study theseexotic states of matter focusing on their application to new computationalparadigms with potentially significant benefit to society.

在这个合作项目中，首席研究员（PI）将研究物质拓扑相的数学模型及其在新兴量子信息领域的应用。 物质拓扑相的例子包括分数量子霍尔液体和拓扑绝缘体：受到极端物理条件（例如，在大磁力下接近零开尔文）的材料似乎表现出拓扑行为。 该项目的主要目标是通过数学模型更好地理解这些奇异物质状态的分类，解决物理和计算驱动的数学问题，并应用拓扑和代数方法来研究它们。例如，可以通过查找与特定模型相关联的表示下的编织组的图像来分析量子计算能力（通用性）的问​​题。此外，现有的物质拓扑态假设模型并不能捕捉到所有的微妙之处（例如费米子拓扑序），因此PI将开发新的模型。固体、液体和气体的经典物质状态有更精细的分类：例如固体晶体可以通过它们的对称性来区分。 新发现的拓扑材料尚未得到充分理解，但有可能用于构建性能优于基于标准微芯片的计算机的（量子）计算设备。最常见的量子计算模型，即量子电路模型，即使不是也需要具有挑战性。由于系统与周围环境的局部相互作用，硬件的准确性不可能具有实用价值。基于奇异物质状态的拓扑模型虽然在数学上更复杂，但对这种相互作用具有固有的不容忍性。 PI 将从数学角度研究这些奇异的物质状态，重点关注它们在新的计算范式中的应用，从而为社会带来潜在的重大利益。

项目成果

Rank-finiteness for modular categories

模块化类别的秩有限性

• DOI: 10.1090/jams/842
• 发表时间: 2013-10-25
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: P. Bruillard;S. Ng;E. Rowell;Zhenghan Wang
• 通讯作者: Zhenghan Wang