基于投影算子的牛顿法求解封闭量子系统的轨道优化问题
A Projection Operator-based Newton Method for the Trajectory Optimization of Closed Quantum Systems
摘要：
量子最优控制是实现快速态制备和门设计的重要技术。在缺乏解析解的情况下，大多数量子最优控制方法依赖于迭代方案来更新解估计。目前，现有的求解器的收敛速度最多是超线性的。本文提出了一种新的通用求解器的量子最优控制的基础上的投影算子牛顿法轨迹优化，或PRONTO。具体而言，所提出的方法使用投影算子将薛定谔方程直接纳入成本函数，然后使用拟牛顿法最小化。在每次迭代中，通过计算线性二次轨迹优化问题的解析解来获得下降方向。由此产生的方法保证单调收敛在每次迭代和二次收敛的解决方案附近。为了强调PRONTO的潜力，我们给出了一个数值示例，使用它来解决量子位的最佳状态到状态映射问题，并将其性能与最先进的二次最优控制方法进行比较。
英文摘要：
Quantum optimal control is an important technology that enables fast state preparation and gate design. In the absence of an analytic solution, most quantum optimal control methods rely on an iterative scheme to update the solution estimate. At present, the convergence rate of existing solvers is at most superlinear. This paper develops a new general purpose solver for quantum optimal control based on the PRojection Operator Newton method for Trajectory Optimization, or PRONTO. Specifically, the proposed approach uses a projection operator to incorporate the Schrödinger equation directly into the cost function, which is then minimized using a quasi-Newton method. At each iteration, the descent direction is obtained by computing the analytic solution to a Linear-Quadratic trajectory optimization problem. The resulting method guarantees monotonic convergence at every iteration and quadratic convergence in proximity of the solution. To highlight the potential of PRONTO, we present an numerical example that employs it to solve the optimal state-to-state mapping problem for a qubit and compares its performance to a state-of-the-art quadratic optimal control method.
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