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）中量子计算普适性的重要问题，探索MBQC与统计力学和凝聚态物理学以及其他量子计算模型的联系。具体来说，阿弗莱克-肯尼迪-利布-田崎（AKLT）模型提供了一个丰富的游乐场，用于探索新的普遍资源状态，并理解量子计算普适性与计算能力中的渗透，空间连通性，磁序和相变的复杂关系。任何二维旋转对称（包括AKLT）哈密顿量的有限谱的存在性的长期未决问题对于通过冷却产生相关资源状态的稳定性也是重要的。这将与分析和数值手段进行研究。该研究还包括寻找新类型的资源状态和开发模型哈密顿量，其热状态可用于量子计算，而无需关闭相互作用。此外，该计划还研究了拓扑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一大类拓扑有序状态。智力优势：MBQC是构建量子计算机的几种模型之一。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开始和执行本地测量的能力。这种实现量子计算机的方法在几个物理系统中很有前途，例如光学晶格中的超冷原子和光子，补充了实现量子计算的其他方法。MBQC还提供了一个概念框架，用于回答量子计算中的基本问题，并连接到其他研究领域。将解决的问题包括：（1）什么纠缠态可以资格作为一个普遍的资源，他们可以出现作为唯一的基态的物理合理的哈密顿？全面的理解可能会在计算能力方面对物质状态进行新的表征。(2)在更高的维度中是否存在广义的Haldom猜想以及如何验证它？解决二维AKLT哈密顿量的谱隙这一长期悬而未决的问题，将使人们深入了解二维中可能的广义哈勒猜想，并为在更高维度中探测各向同性自旋哈密顿量中更丰富的相位铺平道路。(3)拓扑秩序能为探索新的资源状态提供洞察力吗？(4)MBQC模式与其他模式相比有哪些优势？该计划的MBQC研究成果不仅将促进我们对量子计算各个方面及其与凝聚态物理和统计力学思想的联系的认识，而且对未来的量子计算机技术具有潜在的影响。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，并促进斯托尼布鲁克大学的跨学科合作。他将把他对量子计算的研究整合到他目前正在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.