A general formulation of time-optimal quantum control and optimality of singular protocols
A general formulation of time-optimal quantum control and optimality of singular protocols
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DOI：
10.1088/1367-2630/ab8ab3
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发表时间：
2019-12
影响因子：
3.3
通讯作者：
Hiroaki Wakamura;T. Koike
中科院分区：
文献类型：
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作者：
Hiroaki Wakamura;T. Koike
当哈密顿量受任意约束时，我们给出了寻找量子系统时间最优么正演化的一般理论框架。量子臂式计时器(QB)就是这样一个基于变分原理的框架，其缺点是它只处理等式约束。虽然在某些情况下，不等约束可以简化为相等约束，但它们通常不能，特别是当哈密顿量中存在漂移场时，漂移场是不可控的部分。我们首先开发了一个基于庞特里亚金最大值原理(MP)的框架，以处理不平等约束。新框架包含QB作为特例，并给出了它们的详细对应关系。其次，我们讨论了满足MP的奇异控制问题，从而给最优协议的确定带来了困难。为了克服这一困难，我们利用推广的Legendre-Clebsch条件，给出了奇异协议是最优协议的另一个必要条件。第三，我们讨论了漂移、奇异控制和不等式约束之间的一般关系。最后，我们通过一些例子演示了我们的框架和结果是如何工作的。我们还讨论了奇异控制的物理意义。
We present a general theoretical framework for finding the time-optimal unitary evolution of the quantum systems when the Hamiltonian is subject to arbitrary constraints. Quantum brachistochrone (QB) is such a framework based on the variational principle, whose drawback is that it only deals with equality constraints. While inequality constraints can be reduced to equality ones in some situations, they usually cannot, especially when a drift field, an uncontrollable part, is present in the Hamiltonian. We first develop a framework based on Pontryagin’s maximum principle (MP) in order to deal with inequality constraints as well. The new framework contains QB as a special case, and their detailed correspondence is given. Second, we address the problem of singular controls, which satisfy MP trivially so as to cause a trouble in determining the optimal protocol. To overcome this difficulty, we derive an additional necessary condition for a singular protocol to be optimal by applying the generalized Legendre–Clebsch condition. Third, we discuss general relations among the drift, the singular controls, and the inequality constraints. Finally, we demonstrate how our framework and results work in some examples. We also discuss the physical meaning of singular controls.