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

费曼的猜想和劳埃德在《科学》杂志上发表的文章并列放在一起。费曼的猜想是，如果我们可以建立一个可以随意操纵的量子模拟器，由自旋1/2的粒子组成，那么我们就可以根据我们想要模拟的粒子之间的相互作用来设计这些粒子之间的相互作用，从而只需在量子模拟器上执行适当的测量就可以预测物理量的值。劳埃德在《科学》杂志上发表的文章肯定了哈密顿模拟问题及其应用的深刻严重性。为了准备这样的模拟，必须将数学上描述的酉算符转换为可识别为量子电路的酉算符，然而目前的大多数技术容易出现近似误差，从而影响模拟的真实性。该项目通过松散的动力解决了Cartan分解的创新途径中的困难。这一过程不仅在数值上是可行的，而且还产生了真正的么正合成，该合成在精度和所需合成成分的使用方面都是最优的。该项目旨在建立理论和算法基础，并发展数值方法。完成后，理论和实验有望找到从量子模拟延伸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稳健性。这项工作的成果将巩固在一个标准框架下的许多研究。初步结果表明，该项目具有良好的发展潜力。预计至少会有一名研究生就该项目的主题进行培训。在经典计算机上模拟量子系统的时间演化是很困难的--描述一个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，更不用说集成其运动方程了。量子机上的哈密顿模拟是这一挑战的可能解决方案-假设由自旋1/2的粒子组成的量子系统可以随意操纵，那么就可以根据要模拟的粒子来设计这些粒子之间的相互作用，从而只需对系统进行适当的测量就可以预测物理量的值。因此，在数学上描述为逻辑解的酉运算符和可识别为用于执行的量子电路的酉运算符之间建立联系对于算法设计和电路实现至关重要。目前的大多数技术都容易产生近似误差。这个项目是通过松弛动力学的概念来处理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么正综合，它在精度和积分误差可控的情况下都是最优的，并且只使用最少所需的合成部件。本项目的目标是建立以下目标的理论和算法基础：1)开发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系统上测试该算法。将开展六项具体任务，以得出实现这些目标的理论和数值方法。完成后，理论和实验有望从量子模拟延伸到多体问题，以及量子信息科学的各种研究努力。这一奖项反映了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。

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.