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

信息处理设备，如计算机，在现代生活中已经无处不在，不可或缺。一种新的有前途的范式，称为量子信息处理（QIP），利用微观量子变量（如电子的自旋）来编码信息。与传统的非黑即白（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，使我们能够以灰色阴影处理信息，从而获得比传统设备的巨大改进。然而，这种模式存在两个主要挑战。一个是如何在现有的几个由几十个量子比特组成的实验平台上，进一步扩大QIP设备的规模。二是确定QIP设备超越传统计算机的信息处理任务。本课题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量子多体现象，可以用量子信息概念来观察和解释，这是有利的。特别是，最近发现，表现出某些对称性和拓扑现象的强受挫量子自旋系统可以作为量子计算机使用。通过这个具体的例子，本项目通过分析对称、几何和拓扑的重要作用，寻求宏观量子秩序与量子优势在计算和模拟中的深层联系。这项研究也将有助于建立量子信息科学的知识库，并在一个高度跨学科的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有通过测量才能显现出来。从贝尔不等式到最近所谓的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截然不同的成分以达到实际效果。基于测量的量子计算（MBQC）框架便于研究这种相互作用和分析计算中量子加速的来源。最近，人们认识到，在被称为对称保护拓扑序（SPTO）的受挫量子自旋系统的量子自旋液相中自然存在的某些宏观纠缠能够成为MBQC的资源。该项目利用宏观量子顺序和计算复杂性之间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、几何形状和拓扑结构如何增强量子计算和模拟？”考虑到不同的晶格几何和相应的子晶格对称性，本项目将发展SPTO的分类，并分析量子元胞自动机的相关结构。这些特性为某些短深度量子电路利用SPTO的序参量建立了一条通往量子优势的直接途径。该方法将有效解决噪声中尺度量子技术时代的两个关键挑战，即探索SPTO的自然结构，以实现具有良好可扩展性控制的鲁棒宏观纠缠，以及超越可能的经典模拟的复杂计算和模拟。从广义上讲，该项目是量子信息科学和量子多体物理两个研究领域的交叉融合，在量子模拟时代即将到来之际，量子多体物理提出了许多问题，量子计算机应该比传统计算机更有效地解决这些问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.