Quantum Cellular Automata Dynamics: Integrability, Many-Body Decoherence, and Complex Entanglement

复杂性科学是21世纪深刻的突出问题之一，涉及意识的生物学起源等深刻问题。一般来说，复杂性的起源是什么？它已经出现在量子力学中了吗？我们是否需要一个新的物理理论，或者复杂性是我们已知知识的产物？噪声中尺度量子（NISQ）计算机涉及越来越多的相互作用的量子子系统，从而增加了复杂性。在NISQ计算机中研究这种突发性复杂性的一种方法是使用Goldilocks量子元胞自动机（QCA）。Goldilocks QCA是写入量子电路的动态计算规则，涉及在局部邻域内不太多和不太少之间的权衡或平衡-因此称为“Goldilocks”。促进科学进步，该项目将连接以前不同的基础数学和科学概念，如可积性和复杂性；指出了2D QCA可能超越经典计算演示的方式；并通过不可逆的基本QCA为开放量子系统开发新的量子计算范式，利用环境而不是像生命系统那样回避环境。此外，该团队还提出了一个多方面的方法来实现更广泛的影响目标。首先，他们每年将指导2-3名来自美国最大的本科物理项目之一的本科生进行研究。其次，他们将通过在《物理杂志：复杂性》（Journal of Physics: Complexity）的执行编辑委员会任职，参与国际事务；并通过美国国务院作为杰斐逊科学研究员校友支持科学外交。第三，他们将参与混合和虚拟课堂环境下学生协作网络定量研究的教育发展；并在矿业荣誉项目中开设科学外交课程，面向所有STEM学生开放，并向公众传播。第四，他们将通过在该领域取得成功的支持性、多样化的团体环境来增加物理学的多样性、公平性和包容性。最后，目前量子劳动力不足以满足我国量子倡议的要求。该项目将培养大量本科、硕士和博士水平的学生，他们具有近期模拟和数字量子计算机建模的实际动手经验，跨越量子网络，以帮助满足这一迫切需求。在这个关于NISQ设备支持的纠缠动力学的新见解的项目中，团队将追求(i)复杂纠缠，（ii）多体退相干和（iii）可积性之间的联系。QCA提供了一个实用的量子测试平台，在最近的工作中被证明可以在数字量子计算机上实现。(i)他们将系统地研究全套准一维（扩展5位点）和二维QCA，它们的复杂性特性，以及它们在Sycamore芯片上数字量子电路中的有效实现前景。复杂纠缠是可积性的结果吗？或者，恰好可以证明包含1D (3-site) Goldilocks QCA的可积系统，是复杂性的转移焦点吗？这仍有待解决。（ii）他们将在数字量子电路和模拟量子模拟器的现实条件下，系统地处理量子轨迹演化下准1d和2D Goldilocks和非Goldilocks QCA的退相干特性。然后，他们将在先前对16个可逆基本1D QCA的研究的基础上，从小量子系统中剩余的240个不可逆基本1D QCA中选出有代表性的选择，看看多体退相干特性是否成立。这项研究将包括规则110，这是经典的图灵完备；并为实现量子版康威人生游戏的长期目标指明了道路。（iii）完整地证明了一维3点QCA在相当一般的条件下是可积的，并有力地证明了该证明可扩展到任意大小邻域的猜想。为此，他们将通过基于新发现的守恒电荷的截断广义吉布斯系综来研究热化；通过非时序相关器的动态置乱；佩奇曲线和光谱特征，包括许多身体疤痕。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。

Complexity science is one of the deep outstanding problems of the 21st century, tapping into such profound questions as the biological origin of consciousness. What are the origins of complexity generally? Does it appear already in quantum mechanics? Do we require a new physical theory, or is complexity an outgrowth of what we know? Noisy intermediate scale quantum (NISQ) computers involve an increasing number of interacting quantum subsystems giving rise to complexity. One way to study such emergent complexity in NISQ computers is with Goldilocks quantum cellular automata (QCA). Goldilocks QCA are dynamic computational rules written into a quantum circuit which involve a trade-off or balance between not too many and not too few in a local neighborhood -- thus the term "Goldilocks". Enhancing the progress of science, this project will connect formerly disparate foundational mathematical and scientific concepts, such as integrability and complexity; point the way to possible beyond-classical computing demonstrations with 2D QCA; and develop new quantum computing paradigms for open quantum systems via irreversible elementary QCA making use of the environment instead of avoiding it, just as living systems do. Additionally, the team proposes a multi-faceted approach to meet broader impact goals. First, they will mentor 2-3 undergraduates per year in research from one of the largest undergraduate physics programs in the US. Second, they will engage internationally by serving on the executive editorial board of the Journal of Physics: Complexity; and by supporting science diplomacy via the U.S. Department of State as a Jefferson Science Fellow alumni. Third, they will engage in educational development in the quantitative study of student collaboration networks in hybrid and virtual class environments; and by creation of a science diplomacy course in the Mines honors program accessible to all STEM students and publicly disseminated. Fourth, they will increase diversity, equity, and inclusion in physics via a supportive, diverse group environment with a track record of success in this area. Finally, at present the quantum workforce is insufficient to meet our national quantum initiative requirements. This program will train a sizeable cadre of students at BS, MS, and PhD levels with practical hands-on experience modeling near-term analog and digital quantum computers, crossing over into quantum networks, to help fill this pressing need. In this project on new insight into entanglement dynamics enabled by NISQ devices, the team will pursue the connections between (i) complex entanglement, (ii) many-body decoherence, and (iii) integrability. QCA provide a practical quantum test-bed, proven in recent work to be realizable on digital quantum computers. (i) They will systematically study the full set of quasi-1D (extended 5-site) and 2D QCA, their complexity properties, and their prospective efficient implementation in digital quantum circuits on the Sycamore chip. Is complex entanglement a result of integrability? Or, are integrable systems, which happen to provably encompass 1D (3-site) Goldilocks QCA, a red herring for complexity? This remains to be resolved. (ii) They will systematically treat the decoherence properties of quasi-1D and 2D Goldilocks and non-Goldilocks QCA under quantum trajectories evolution with realistic conditions for both digital quantum circuits and analog quantum simulators. They will then build on prior studies of the 16 reversible elementary 1D QCA to a representative selection of the remaining 240 irreversible elementary 1D QCA in small quantum systems to see if many-body decoherence properties hold up. This study will include Rule 110, which is classically Turing complete; and point the way to the long-term goal of realizing a quantum version of Conway's game of life. (iii) They will provide a complete proof that 1D 3-site QCA under quite general conditions are integrable, and strong evidence for a conjecture that the proof extends to arbitrary sized neighborhoods. To this end they will study thermalization via a truncated generalized Gibbs ensemble based on newly discovered conserved charges; scrambling via the dynamics of the out-of-time-order correlator; and Page curves and spectral characteristics including many-body scars.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.