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

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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.