Weaving Stability from Dissipation: Fixed-Point Engineering for Quantum Information Processing

理解物理系统如何接近平衡是统计物理学中的一个中心问题，当系统按照量子力学定律运行时，这个问题就更加深刻。作为一个熟悉的例子，想想一杯热茶或一杯冷水是如何平衡到室温的。一个变冷，另一个变暖；但在每种情况下，单独的分子相互作用都会影响这些系统接近平衡的速度和波动。一个更微妙的例子来自量子计算领域，在量子计算领域，信息可以编码在单个分子的量子力学自旋态中。这样做的一个令人信服的动机是生产功能更强大的计算机，其内存和信息处理能力远远优于当今的经典计算机。然而，由于趋于平衡的弛豫会破坏存储在分子中的量子信息，平衡通常被认为是量子计算的障碍或障碍。这个项目的目标是找到利用平衡态的稳定性来改进量子计算协议的方法。将发展表征和影响多粒子量子态稳定性的方法。我们将探索如何将任何给定的初始状态“吸引”或推向一组稳定的平衡态，这些平衡态包含量子计算所需的真正的非经典多粒子关联(“纠缠”)。这项研究将解决在什么条件下可以实现稳定到期望的量子态的基本问题，它将扩展描述和控制耗散量子系统的工具包和方法。该项目还为从事量子信息科学、量子控制理论、应用数学、多体和统计物理之间跨学科边界的学生提供教育和培训。在过去的几年里，首席研究者已经建立了严格的条件，使感兴趣的一般量子态成为一类受现实局部性约束的连续时间无挫折性马尔可夫动力学的唯一不动点。在这些结果的基础上，本项目旨在探索在很大程度上尚不清楚的物理环境中的量子稳定化问题--例如：(1)比局部性施加的更一般约束下的连续时间马尔可夫动力学，包括对扰动的敏感性和周期性时变的Floquet-马尔可夫生成器；(2)受约束的离散时间马尔可夫动力学，其中出现了在有限时间内精确耗散态准备和耗散编码的有趣可能性；(3)随机开放系统动力学，它可能有助于揭示一般(随机)目标状态或子空间的可镇定性。所使用的方法将是分析和数值两种方法，并将利用从应用数学和量子控制理论(包括算子代数、Lyapunov稳定性技术和半定规划)到量子信息理论和多体物理(特别是纠缠理论、信息保持结构和张量网络技术)的各种工具。从理论上讲，一个中心主题将是澄清多体和统计力学的核心特征--特别是平衡态下量子关联的复杂性、没有挫折感、基础哈密顿量的空隙性质或交换性--可能在多大程度上影响量子信息处理稳定任务的可行性和效率。从实用的角度来看，这些研究的目标是改进耗散量子态准备和量子信息编码的协议。

Understanding how physical systems approach equilibrium is a central question in statistical physics, and this topic is all the more profound when systems behave according to the laws of quantum mechanics. As a familiar example, consider how a hot cup of tea or a cold glass of water equilibrate to room temperature. One cools down while the other warms up; but in each case individual molecular interactions affect the rate and the fluctuations with which these systems approach equilibrium. A more nuanced example comes from the field of quantum computing, where information can be encoded in the quantum mechanical spin states of individual molecules. One compelling motivation for this is to produce more powerful computers with memory and information processing capabilities far superior to today's classical computers. However, since relaxation towards equilibrium can disorganize the quantum information stored in molecules, equilibration is normally considered an obstacle or a hindrance for quantum computing. The goal of this project is to find ways to utilize the stability of states near equilibrium in order to improve quantum computing protocols. Methods to characterize and influence the stability of multi-particle quantum states will be developed. Techniques to "attract" or nudge any given initial state towards a set of stable equilibrium states that contain genuinely non-classical multi-particle correlations ("entanglement") needed for quantum computing will be explored. This research will address fundamental questions about the conditions under which stabilization to a desired quantum state may be achieved, and it will expand the toolkit and of methods to characterize and control dissipative quantum systems. This project also provides education and training for students working on subjects at the cross-disciplinary boundaries between quantum information science, quantum control theory, applied mathematics, and many-body and statistical physics. Over the last few years, the principal investigator has established rigorous conditions for a general quantum state of interest to be the unique fixed point of a class of continuous-time "frustration-free" Markovian dynamics, subject to realistic locality constraints. Building on these results, this project aims to explore quantum stabilization problems in physical settings that remain largely uncharted as yet -- for instance: (i) Continuous-time Markovian dynamics under more general constraints than imposed by locality, including sensitivity to perturbations and periodically time-varying Floquet-Markov generators; (ii) Constrained discrete-time Markovian dynamics, for which the intriguing possibilities of exact dissipative state preparation and dissipative encoding in finite time arise; (iii) Randomized open-system dynamics, which may shed light on the stabilizability properties of generic (random) target states or subspaces. The methods to be employed will be both analytical and numerical, and will draw on a diverse range of tools from applied mathematics and quantum control theory (including operator algebras, Lyapunov stability techniques, and semi-definite programming), to quantum information theory and many-body physics (notably, entanglement theory, information-preserving structures, and tensor-network techniques). Theoretically, a central theme will be to clarify the extent to which core features from many-body and statistical mechanics -- in particular, the complexity of quantum correlations in the equilibrium state, the lack of frustration, the gapped nature or the commutativity of the underlying "parent" Hamiltonians -- may be brought to bear on the feasibility and efficiency of stabilization tasks for quantum information processing. From a practical standpoint, the goal of these investigations is to improve protocols for dissipative quantum state preparation and quantum information encoding.