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：本文提出了比经典算法呈指数级加速的高效量子算法，用于求解量子最优控制问题。该问题涉及在系统由含时薛定谔方程支配的情况下，寻找使物理量在时间$T$处最大化的控制变量。此类控制问题与机器学习存在密切联系。我们的算法基于含时哈密顿量模拟方法和快速梯度估计算法。同时，我们提供了全面的误差分析，量化了来自各步骤（如控制函数的有限维表示、薛定谔方程的离散化、数值求积和优化）的总误差。我们的量子算法需要容错量子计算机的支持。