Quantum Walks and Cellular Automata for Quantum Information Processing

量子场论是我们最深刻的自然理论的基础，包括粒子物理的标准模型及其扩展。这些理论将量子力学和狭义相对论结合起来，在理解基础物理学方面取得了非常成功的成就。然而，它们有许多数学挑战和开放问题，这使得它们很难用封闭的形式解决，而且它们也很难用计算机模拟。这项拟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)之间的一种有趣的关系，这表明量子场论可能作为称为量子细胞自动机的模型的低能极限而产生，给出了一种不同的方法来推导量子场论。这种关系引发了许多非常有趣的问题。这个项目将为研究生提供博士研究的主题，并在南加州大学发展研究社区和量子信息处理课程方面发挥作用。PI将为本科生开发教材，并开设量子信息科学硕士学位课程，旨在为越来越多从事量子计算机工作的公司服务。量子漫步是经典随机漫步的量子力学类似物。在随机漫游中，粒子沿着图形的边以离散的步长移动，随机选择在每一步采取哪条边。量子行走有类似的数学描述，但它们可以将不同的运动叠加在一起，每一步都是么正变换，而不是随机移动。这种差异导致了与随机行走截然不同的行为，包括干扰效应、波状传播和一系列丰富的其他量子现象。量子行走与量子算法有联系，如搜索和元素区分性；它们还形成了基本物理系统的模型。量子细胞自动机是量子行走的自然多体推广，图中的顶点成为局域量子系统，通过与邻近站点的局部相互作用以离散的时间步长进化。这个项目将探索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；在量子计算机(可以有效地模拟量子行走和量子细胞自动机)上的潜在应用；旋转对称性和洛伦兹不变性等连续对称性如何作为离散理论的极限出现；局部进化、费米统计和正能量之间的紧张；以及作为量子计算的替代模型的多粒子量子行走。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.