求解量子系统时间最优控制问题的同伦算法研究

摘要

This paper presents a homotopic algorithm for time optimal control of quantum systems in order to avoid large computational burdens when we numerically solve the same problem by using a class of gradient algorithm. By introducing a homotopy variable, our algorithm searches solutions along a direction for decreasing the final time, which is different from existing algorithms. In this algorithm, a freedom function is introduced to construct the gradient of a fidelity with respect to input fields. Also, we adopt a direction function to lead the searching direction of the algorithm so as to speed up the algorithm. We apply it into the time-optimal control of state transitions and unitary gate transformations for two quantum systems. Simulation results show the effectiveness of our algorithm.%针对数值求解量子系统时间最优控制问题中反复调用梯度算法导致计算量大的问题,本文提出一类同伦算法用以快速求解量子系统的时间最优控制问题.与已有算法不同,这一算法通过引入同伦变量在减小终端时间的方向上搜索最优解.在这一算法中,可通过自由函数构造保真度函数对控制变量的梯度方向,也可通过方向函数引导算法的搜索方向,以加快算法的搜索速度.本文将这一算法用于求解量子系统态转移和门变换的时间最优控制问题.仿真结果表明这一算法的有效性.