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

在这个合作项目中，主要研究人员（PI）将研究物质拓扑相的数学模型及其在量子信息新兴领域的应用。 物质拓扑相的例子包括分数量子霍尔液体和拓扑绝缘体：材料受到极端的物理条件（例如接近零开尔文，在大磁力下），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，量子计算能力（普适性）的问题可以通过找到与特定模型相关的辫子群的图像来分析。此外，现存的拓扑状态的假设模型不能捕捉所有的微妙之处（如费米拓扑顺序），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。 新发现的拓扑材料还没有被完全理解，但有可能被用来建造（量子）计算设备，性能超过标准的基于微芯片的计算机。最常见的量子计算模型，量子电路模型，由于系统与周围环境的局部相互作用，需要具有挑战性（如果不是不可能的话）的硬件精度才有实用价值。基于物质奇异态的拓扑模型，虽然在数学上更加复杂，但对这种相互作用有一种内在的不容忍。PI将从数学上研究这些奇异的物质状态，重点是将其应用于对社会有潜在重大利益的新计算范式。

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.