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

费曼猜想的并置，“如果我们可以建立一个量子模拟器，由我们可以随意操纵的自旋1/2粒子组成，那么我们将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》杂志上发表的文章肯定了“量子计算机可以被编程来模拟任何局部量子系统”，这表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，必须将数学描述的酉算子转换为可识别为量子电路的酉算子，然而大多数当前技术容易产生影响模拟真实性的近似误差。这个项目解决了困难，从一个创新的途径的Cartan分解通过拉克斯动力学。这个过程不仅在数值上是可行的，而且还产生了一个真正的酉合成，这是最佳的精度和最低限度所需的合成组件的使用。该项目旨在建立理论和算法基础，并开发数值方法。完成后，理论和实验有望从量子模拟扩展到其他领域，如衡量其他方法的质量或评估给定系统的鲁棒性。这项工作的成果将把许多研究巩固在一个标准框架下。初步结果显示了该项目的潜力。在经典计算机上模拟量子系统的时间演化是困难的--甚至描述量子系统所需的计算能力都与其组成部分的数量成指数关系，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方案-假设由自旋1/2粒子组成的量子系统可以随意操纵，那么根据要模拟的粒子来设计这些粒子之间的相互作用是可行的，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是必不可少的。大多数当前的技术都倾向于近似误差。该项目是通过Lax动力学的概念来解决Cartan分解，这不仅在数值上是可行的，而且还产生了一个真正的酉合成，该合成在具有可控积分误差的精度和仅使用最少需要的合成组件方面都是最佳的。该项目旨在为以下目标建立理论和算法基础：1）利用Hamilton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系统上实验该算法。 六个具体的任务将进行推导的理论和数值方法，以达到这些目标。完成后，理论和实验有望从量子模拟扩展到多体问题，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，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。

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.