Symmetry, Geometry, and Topology of Quantum Many-Body States for Quantum Computation

信息处理设备，如计算机，在现代生活中已经变得无处不在和不可或缺。一个新的有前途的范例，称为量子信息处理（QIP），利用微观量子变量（如电子自旋）来编码信息。与传统的黑或白色（所谓的0或1）逻辑相比，反直觉的量子效应，如叠加和量子相关或纠缠，使我们能够处理带有灰色阴影的信息，从而实现对传统设备的巨大改进。然而，这一模式面临两大挑战。一个是弄清楚如何扩大QIP设备的规模，建立在目前由数十个量子比特组成的几个实验平台上。另一个是确定QIP设备超越传统计算机的信息处理任务。本项目的目标是从宏观系统的量子多体理论的角度来解决这些关键问题。例如，认识到超导性和磁性是可以使用量子信息概念来观察和解释的量子多体现象是有利的。特别是，最近发现，表现出一定对称性和拓扑现象的强挫量子自旋系统可能起量子计算机的作用。通过这个具体的例子，该项目通过分析对称性、几何和拓扑的重要作用，寻求宏观量子秩序与计算和模拟中的量子优势之间的深层联系。这项研究还将有助于量子信息科学的知识基础，并有助于在高度跨学科领域培养未来的科学家。纠缠和测量之间的基本相互作用是量子信息科学的核心。虽然纠缠的复杂性代表了一种独特的量子资源，但其特有的非经典特征只能通过测量来揭示。从贝尔不等式到最近对所谓玻色子采样问题的量子模拟，量子信息科学的里程碑式成果都依赖于平衡这两种对立的成分以实现实际效果。基于测量的量子计算（MBQC）框架便于研究这种相互作用，并分析计算中量子加速的起源。最近，人们已经认识到，某些宏观纠缠，这将自然地发现在受抑的量子自旋系统的量子自旋液相中被称为量子保护拓扑序（SPTO），是能够成为MBQC的资源。该项目利用宏观量子序与计算复杂性之间的这种独特而具体的联系，来回答一个关键问题：“SPTO中体现的对称性、几何和拓扑如何赋予量子计算和模拟能力？“考虑到不同的晶格几何形状和相应的子晶格对称性，该项目将开发SPTO的分类并分析量子细胞自动机的相关结构。这些特征建立了一个直接的路径，量子优势使用的序参数SPTO某些短深度量子电路。该方法将有效地解决噪声中等规模量子技术时代的两个关键挑战，因为SPTO的自然结构被探索以实现鲁棒的宏观纠缠，具有良好的可扩展控制以及复杂的计算和模拟，超出了可能的经典模拟。从广义上讲，该项目交叉了两个研究领域，量子信息科学和量子多体物理学，在即将到来的量子模拟时代，当量子多-身体物理学提出了量子计算机应该比传统计算机更有效地解决的许多问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持审查标准。

Information processing devices, like computers, have become ubiquitous and indispensable in modern life. A new promising paradigm, called quantum information processing (QIP), takes advantage of microscopic quantum variables (such as spins of electrons) to encode information. Counterintuitive quantum effects, such as superposition and quantum correlation or entanglement, enable us to process information with shades of gray, as compared with conventional black-or-white (so-called 0-or-1) logic, and so to attain drastic improvements over conventional devices. However, there are two major challenges to this paradigm. One is to figure out how to scale up further QIP devices, building on several current experimental platforms made of dozens of quantum bits. The other is to identify the information processing tasks for which QIP devices surpass conventional computers. The goal of this project is to address these key issues from the perspective of quantum many-body theory of macroscopic systems. It is advantageous, for example, to recognize that superconductivity and magnetism are quantum many-body phenomena that can be viewed and interpreted using quantum information concepts. In particular, it has been recently discovered that strongly frustrated quantum spin systems which manifest certain symmetries and topological phenomena might function as a quantum computer. Through this concrete example, the project seeks a deep connection between macroscopic quantum orders and quantum advantage in computation and simulation, by analyzing the important roles of symmetry, geometry, and topology. This research will also contribute to the knowledge base of quantum information science and to the training of future scientists in a highly interdisciplinary field. A fundamental interplay between entanglement and measurement lies at the heart of quantum information science. While the complexity of entanglement represents a uniquely quantum resource, its characteristic nonclassical features only reveal themselves through measurement. From Bell's inequality to recent quantum simulations of the so-called boson-sampling problem, landmark results of quantum information science have all relied upon balancing these two contrasting ingredients to practical effect. The framework of measurement-based quantum computation (MBQC) is convenient to study such an interplay and to analyze the origin of quantum speed-up in computation. Recently, it has been recognized that certain macroscopic entanglement, which would be naturally found in quantum spin liquid phases of frustrated quantum spin systems called symmetry-protected topological orders (SPTO), is capable of becoming a resource for MBQC. The project takes advantage of this unique, concrete connection between macroscopic quantum orders and computational complexity, to answer a key question "How do symmetry, geometry, and topology embodied in SPTO empower quantum computation and simulation?" Considering different lattice geometries and corresponding sublattice symmetries, the project will develop the classification of SPTO and analyze the associated structure of quantum cellular automata. These characterizations establish a direct route to quantum advantage using the order parameters of SPTO for certain short-depth quantum circuits. The approach would be effective to tackle two key challenges in the era of noisy intermediate-scale quantum technology, in that the natural structure of SPTO is explored to realize robust macroscopic entanglement with well-scalable control as well as complex computation and simulation beyond possible classical simulation. Broadly, the project cross-fertilizes two research fields, quantum information science and quantum many-body physics, timely at the coming age of quantum simulation when quantum many-body physics suggests many problems which quantum computers should be more efficient to solve than conventional computers.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.