Quantum Walks and Cellular Automata for Quantum Information Processing

量子场论是我们关于自然的最深层理论的基础，包括粒子物理学的标准模型及其扩展。这些理论结合了量子力学和狭义相对论，在理解基础物理学方面非常成功。然而，它们有许多数学挑战和开放问题，这使得它们很难以封闭的形式解决，而且它们也很难用计算机模拟。这项提议的研究探索了量子行走和相对论量子方程（如狄拉克方程）之间的有趣关系，这表明量子场论可能出现在称为量子细胞自动机的低能量极限模型中，为推导量子场论提供了一种不同的方法。这种关系提出了许多非常有趣的问题。该项目将为研究生提供博士研究课题，并在南加州大学发展研究社区和量子信息处理课程中发挥作用。PI将为量子信息科学的本科生和硕士学位课程开发教育材料，旨在为越来越多的从事量子计算机工作的公司服务。量子行走是经典随机行走的量子力学类比。在随机漫步中，粒子沿着图的边缘以离散的步骤移动，每一步随机选择哪条边。量子行走也有类似的数学描述，但它们不是随机移动，而是不同移动的叠加，每一步都是一个幺正变换。这种差异导致了与随机漫步截然不同的行为，具有干涉效应、波状传播和丰富的其他量子现象。量子行走与量子算法有联系，如搜索和元素独特性；它们也构成了基本物理系统的模型。量子元胞自动机是量子行走的自然多体推广，其中图的顶点成为局部量子系统，通过与邻近点的局部相互作用以离散时间步进化。本项目将探讨量子行走和量子细胞自动机的几个方面：作为相对论量子波函数和量子场论的离散模型；量子计算机模拟的潜在应用（可以有效地模拟量子行走和量子细胞自动机）；连续对称，如旋转对称和洛伦兹不变性如何作为一个离散理论的极限出现；局部演化、费米统计和正能量之间的紧张关系；多粒子量子是量子计算的另一种模型。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。

Quantum field theories underlie our deepest theories of nature, including the standard model of particle physics and its extensions. These theories unite quantum mechanics and special relativity, and have been very successful in understanding fundamental physics. However, they have many mathematical challenges and open questions that make them hard to solve in closed form, and they can also be very difficult to simulate with computers. This proposed research explores an intriguing relationship between quantum walks and relativistic quantum equations (like the Dirac equation), which suggests that quantum field theories may arise as low-energy limits of models called quantum cellular automata, giving a different approach to deriving quantum field theories. This relationship raises many very interesting questions. This project will provide topics for Ph.D. research for graduate students, and play a role in developing the research community, and courses in quantum information processing, at USC. The PI will develop educational materials for undergraduates and a master's degree program in Quantum Information Science which aims to serve the growing number of companies working on quantum computers.Quantum walks are quantum-mechanical analogues of classical random walks. In random walks, particles move in discrete steps along the edges of graphs, choosing randomly which edge to take at each step. Quantum walks have a similar mathematical description, but instead of moving randomly they can make superpositions of different moves, with each step being a unitary transformation. This difference leads to dramatically different behavior than random walks, with interference effects, wave-like propagation and a rich array of other quantum phenomena. Quantum walks have connections to quantum algorithms, such as search and element distinctness; they also form models for fundamental physical systems. Quantum cellular automata are a natural many-body generalization of quantum walks, where the vertices of the graph become localized quantum systems, evolving in discrete time steps by interacting locally with neighboring sites. This project will explore several aspects of quantum walks and quantum cellular automata: as discrete models of relativistic quantum wave functions and quantum field theories; potential applications to simulations on quantum computers (which can simulate quantum walks and quantum cellular automata efficiently); how continuous symmetries such as rotational symmetry and Lorentz invariance can arise as a limit of a discrete theory; the tension between local evolution, Fermi statistics, and positive energies; and multiparticle quantum walks as an alternative model of quantum computation.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.