In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time $T$, where the system is governed by a time-dependent Schr\"odinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schr\"odinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.

翻译：在这篇论文中，我们提出了高效的量子算法，其运行速度比经典算法快指数级，用于解决量子优化控制问题。这个问题涉及到在时间$T$内，由时间相关的薛定谔方程控制的系统中寻找最大化物理量的控制变量。这种控制问题还与机器学习有着密切的关系。我们的算法基于一种时间相关的哈密顿模拟方法和一个快速梯度估计算法。我们还提供了全面的误差分析，以量化来自各种步骤的总误差，例如控制功能的有限维表示，薛定谔方程的离散化，数值积分和优化等。我们的量子算法需要容错的量子计算机。