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.

Feynman的概念的并置，“如果我们可以构建一个量子模拟器，由我们可以随意操纵的自旋 - ½粒子组成，那么我们将能够根据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，并通过简单地对物理数量进行量子的数量进行预测，并通过对量子进行适当的测量来进行量子量的计算，并计算了量子的计算，并计算了量子的计算，并在计算机上进行了计算。可以编程以模拟任何局部量子系统。为了准备这样的模拟，必须将数学描述的单一操作员转换为统一操作员将其识别为量子电路，但是大多数当前技术都容易出现影响模拟真实性的近似错误。该项目解决了通过lax Dynamics的创新途径的创新大道的困难。这个过程不仅基本上是可行的，而且还会产生真正的单一合成，在精确和使用最少所需的合成组件方面都是最佳的。该项目旨在建立理论和算法基础并开发数值方法。完成后，预计该理论和实验将发现适用性从量子模拟扩展到其他领域，例如测量其他方法的质量或评估给定系统的鲁棒性。这项工作的收益将在一个标准框架下巩固许多研究。初步结果表明该项目的潜在潜力。预计对至少一名研究生对项目主题进行培训。为了模拟经典计算机上量子系统的时间演变，这很难 - 甚至用其构成的数量成倍地描述量子系统规模所需的计算能力，更不用说集成了其运动方程。量子机上的Hamiltonian模拟是解决这一挑战的可能解决方案 - 假设可以随意操纵自旋 - ½颗粒组成的量子系统，那么可以根据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，从而通过简单地在系统上执行适当的测量值来预测物理量的值。因此，在数学上描述为逻辑解决方案的统一操作员之间建立了联系，而单一操作员则将其识别为执行的量子电路，因此对于算法设计和电路实现至关重要。当前大多数技术容易出现近似错误。该项目是通过LAX动力学的概念来应对Cartan的分解，这不仅是可行的，而且还会产生真正的单一合成，这在受控的积分误差和仅使用最小必需的合成组件的精度方面都是最佳的。该项目旨在建立目标的理论和算法基础：1）利用汉密尔顿亚甲虫的几何特性； 2）描述一种得出松弛动力学的常见机制； 3）开发基于分解的量子算法； 4）在IBM量子集线器系统上实验算法。将执行六项特定任务，以得出实现这些目标的理论和数值方法。完成后，该理论和实验有望找到从量子模拟扩展到多体问题的适用性，并在量子信息科学领域的各种研究努力上。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响来评估的珍贵的支​​持。