Preparing Hamiltonians for Quantum Simulation: A Computational Framework for Cartan Decomposition via Lax Dynamics

为量子模拟准备哈密顿量：通过 Lax 动力学进行嘉当分解的计算框架

• 批准号: 2309376
• 负责人: Moody Chu
• 金额: $ 40万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309376&HistoricalAwards=false
• 关键词: Preparing; Hamiltonians; Quantum; Simulation; Computational

The juxtaposition of Feynman's conjecture that, “if we could build a quantum simulator at our disposal, composed of spin-½ particles that we could manipulate at will, then we would be able to engineer the interaction between those particles according to the one we want to simulate, and thus predict the value of physical quantities by simply performing the appropriate measurements on the quantum simulator,” and Lloyd's article in Science affirming that, “quantum computers can be programmed to simulate any local quantum system,” evinces the profound gravity of the Hamiltonian simulation problem and its applications. To prepare for such a simulation, it is essential to convert the unitary operators described mathematically to the unitary operators recognizable as quantum circuits, yet most current techniques are prone to approximation errors which affect the simulation authenticity. This project tackles the difficulties from an innovative avenue of the Cartan decomposition via the Lax dynamics. Not only is this process numerically feasible, but also produces a genuine unitary synthesis that is optimal in both the precision and the usage of minimally required synthesis components. This project aims to establish theoretic and algorithmic foundations and develop numerical methods. Upon completion, the theory and the experiments are expected to find applicability extending from quantum simulation to other areas such as gauging the quality of other approaches or evaluating the robustness of a given system. The gains from this work will solidify the many studies under one standard framework. Preliminary results show promising potential of this project. Training of at least one graduate student on the topics of the project is expected.To simulate the time evolution of a quantum system on a classical computer is hard - The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge - Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution is therefore essential for algorithm design and circuit implementation. Most current techniques are prone to approximation errors. This project is to tackle the Cartan decomposition via the notion of Lax dynamics, which not only is numerically feasible, but also produces a genuine unitary synthesis that is optimal in both the precision with controllable integration errors and the usage of only minimally required synthesis components. This project aims at establishing theoretic and algorithmic foundations of the goals: 1) exploit the geometric properties of Hamiltonian subalgebras; 2) describe a common mechanism for deriving the Lax dynamics; 3) develop a decomposition-based quantum algorithm; and 4) experiment the algorithm on the IBM Quantum Hub systems. Six specific tasks will be undertaken to derive the theory and numerical methods to reach these goals. Upon completion, the theory and the experiments are expected to find applicability extending from quantum simulation to many-body problem, and to various research endeavors in quantum information science.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.

费曼的猜想是：“如果我们能够构建一个由我们可以随意操纵的自旋 1/2 粒子组成的量子模拟器，那么我们就能够根据我们想要模拟的粒子来设计这些粒子之间的相互作用，从而通过简单地在量子模拟器上进行适当的测量来预测物理量的值”，劳埃德在《科学》杂志上的文章肯定了这一点，“量子 计算机可以被编程来模拟任何局部量子系统，”证明了哈密顿模拟问题及其应用的深刻重要性。为了准备这样的模拟，必须将数学上描述的酉算子转换为可识别为量子电路的酉算子，但大多数现有技术容易出现近似误差，从而影响模拟的真实性。该项目通过松散动力学解决了嘉当分解的创新途径的困难。该过程不仅在数值上可行，而且还产生真正的单一合成，该合成在精度和最少所需合成组件的使用方面都是最佳的。该项目旨在建立理论和算法基础并开发数值方法。完成后，该理论和实验预计将找到从量子模拟扩展到其他领域的适用性，例如衡量其他方法的质量或评估给定系统的鲁棒性。这项工作的成果将巩固一个标准框架下的许多研究。初步结果显示该项目具有良好的潜力。预计将至少对一名研究生进行有关该项目主题的培训。 在经典计算机上模拟量子系统的时间演化是很困难的——描述量子系统所需的计算能力与其组成部分的数量呈指数级增长，更不用说积分其运动方程了。量子机器上的哈密顿模拟是解决这一挑战的一个可能的解决方案——假设一个由自旋 1/2 粒子组成的量子系统可以随意操纵，那么可以根据要模拟的粒子来设计这些粒子之间的相互作用，从而通过简单地对系统进行适当的测量来预测物理量的值。因此，在数学上描述为逻辑解决方案的酉算子与可识别为用于执行的量子电路的酉算子之间建立联系对于算法设计和电路实现至关重要。大多数当前技术都容易出现近似误差。该项目旨在通过 Lax 动力学的概念来解决嘉当分解问题，这不仅在数值上可行，而且还产生真正的酉综合，该综合在具有可控积分误差的精度和仅使用最少所需的综合组件方面都是最佳的。该项目旨在建立目标的理论和算法基础：1）利用哈密顿子代数的几何性质； 2) 描述导出 Lax 动力学的通用机制； 3）开发基于分解的量子算法； 4) 在 IBM Quantum Hub 系统上试验该算法。 将开展六项具体任务来推导出实现这些目标的理论和数值方法。完成后，该理论和实验预计将适用于从量子模拟到多体问题以及量子信息科学的各种研究工作。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。