Aspects of Quantum Computational Universality in the Measurement-Based Models

本研究将探讨基于测量的量子计算（MBQC）中量子计算通用性的重要问题，探索MBQC与统计力学和凝聚态物理思想以及其他量子计算模型的联系。具体来说，Affleck-Kennedy-Lieb-Tasaki （AKLT）模型为探索新的通用资源状态和理解量子计算普适与渗透、空间连通性、磁序和计算能力相变之间的复杂关系提供了丰富的平台。任何二维旋转对称（包括AKLT）哈密顿量的有限谱的存在性这一长期悬而未决的问题对于通过冷却产生相关资源态的稳定性也很重要。这将用解析和数值方法来研究。该研究还包括寻找新型的资源状态和开发模型哈密顿量，其热态可以用于量子计算，而无需关闭相互作用。此外，本程序还研究了拓扑有序如何用于量子计算，以及MBQC如何提供一种有效的方法来创建大类拓扑有序状态。智力优势：MBQC是构建量子计算机的几个模型之一。从本质上讲，所需要的只是一个合适的高度纠缠的资源状态，以及执行局部测量的能力。这种实现量子计算机的方法在几个物理系统中很有前途，例如光学晶格和光子中的超冷原子，补充了实现量子计算的其他方法。MBQC还为回答量子计算中的基本问题和连接其他研究领域提供了一个概念框架。将要解决的问题包括：(1)什么样的纠缠态可以作为一种普遍的资源，它们可以作为物理上合理的哈密顿量的唯一基态出现吗？一个完整的理解可能会导致在计算能力方面对物质状态的新表征。(2)高维是否存在广义霍尔丹猜想，如何检验？解决二维AKLT哈密顿谱间隙这一长期悬而未决的问题，将有助于深入了解二维中可能的广义霍尔丹猜想，并为探索高维各向同性自旋哈密顿谱中更丰富的相铺平道路。(3)拓扑顺序能否为寻找新的资源状态提供洞见？(4) MBQC模式相对于其他模式是否有优势？MBQC在这个项目中的研究成果，不仅会提高我们对量子计算各个方面的认识，以及它与凝聚态物理和统计力学思想的联系，而且对未来的量子计算机技术也有潜在的影响。更广泛的影响：PI正在主动组织一个论坛，讨论量子信息科学的科学成果，并促进石溪大学的跨学科合作。他将把他在量子计算方面的研究整合到他目前和即将为本科生和研究生开发的课程中。该项目还将包括培训一名研究生和指导一名博士后研究人员。

This research will investigate important issues of quantum computational universality in measurement-based quantum computation (MBQC), explore connections of MBQC to ideas in statistical mechanics and condensed matter physics, and other quantum computational models. Specifically, the Affleck-Kennedy-Lieb-Tasaki (AKLT) models supply a rich playground for exploring new universal resource states and for understanding the intricate relations of quantum computational universality to percolation, spatial connectivity, magnetic order, and phase transitions in computational power. The long standing open question of the existence of a finite spectral of any two-dimensional rotationally symmetric (including AKLT) Hamiltonians is important also for the stability of generating related resource states by cooling. This will be studied with both analytic and numerical means. The research also includes searching for new types of resource states and developing model Hamiltonians whose thermal states can be used for quantum computation without the need to switch off interactions. Furthermore, this program studies how topological order can be of use to quantum computation, and conversely, how MBQC offers an efficient means to create a large class of topologically ordered states. Intellectual Merit: MBQC is one of the several models for building quantum computers. Essentially, all that is needed is a suitable highly entangled resource state to begin with and the ability to perform local measurements. This approach of realizing a quantum computer is promising with several physical systems, such as ultracold atoms in optical lattices and photons, complementing other approaches of implementing quantum computation. MBQC also provides a conceptual framework for answering fundamental questions in quantum computation and for bridging to other areas of research. The questions that will be addressed include: (1) What entangled states can qualify as an universal resource and can they arise as unique ground states of physically reasonable Hamiltonians? A complete understanding may lead to novel characterization of states of matter in terms of computational capability. (2) Is there a generalized Haldane conjecture in higher dimensions and how to test it? Tackling the long standing open question of the spectral gaps of two-dimensional AKLT Hamiltonians will give insight to a possible generalized Haldane conjecture in 2D and pave the road for probing richer phases in isotropic spin Hamiltonians in higher dimensions. (3) Can topological order provide insight to the quest of new resource states? (4) Are there advantages over others that the MBQC model offers? The research findings of MBQC from this program will not only advance our knowledge on various aspects of quantum computation and its connection to ideas in condensed matter physics and statistical mechanics, but also have potential impact on future quantum computer technology. Broader Impacts : The PI is taking the initiative in organizing a forum for discussing scientific results in quantum information science and stimulating collaboration across disciplines at Stony Brook University. He will integrate his research on quantum computation in the courses that he is currently and will be developing for both undergraduate and graduate students. This project will also include training of a graduate student and mentoring of a postoctoral researcher.