探索量子控制优化轨迹的复杂性
Exploring the complexity of quantum control optimization trajectories
摘要：
量子系统动力学的控制通常通过寻找合适的施加场来执行。作为场的函数的物理目标形成了量子控制景观，其拓扑结构在某些条件下已被证明不包含临界点次优陷阱，从而能够有效地搜索给出目标的全局最大值的场。本文论述了结构的景观作为一种补充，拓扑临界点功能。最近的工作表明，景观结构是非常有利的状态到状态的转移概率的优化，在基于梯度的控制轨迹的全局最大值几乎是直线路径。景观结构被编码在度量$R\geq 1.0$中，定义为控制轨迹的长度与初始控制和最优控制之间的欧几里得距离的比率。$R=1$的值将指示到最佳可观测值的精确直线轨迹。本文将态-态跃迁几率的结果推广到量子系综和幺正变换控制的情形。同样，几乎直线的轨迹占主导地位，我们证明，$R$可以采取的值接近1.0的高精度。然而，优化轨迹与临界鞍子流形的相互作用被发现会影响景观结构。一个基本的关系所需的完全直的基于梯度的控制轨迹推导出其中的量子控制景观的梯度必须是一个本征函数的海森。这种关系是景观结构的指标，并可以提供一种手段，以确定物理条件时，控制轨迹可以实现完美的线性。集体有利的景观拓扑和结构为理解为什么可以容易地实现最佳量子控制提供了基础。
英文摘要：
The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transition probabilities, in that gradient-based control trajectories to the global maximum value are nearly straight paths. The landscape structure is codified in the metric $R\geq 1.0$, defined as the ratio of the length of the control trajectory to the Euclidean distance between the initial and optimal controls. A value of $R=1$ would indicate an exactly straight trajectory to the optimal observable value. This paper extends the state-to-state transition probability results to the quantum ensemble and unitary transformation control landscapes. Again, nearly straight trajectories predominate, and we demonstrate that $R$ can take values approaching 1.0 with high precision. However, the interplay of optimization trajectories with critical saddle submanifolds is found to influence landscape structure. A fundamental relationship necessary for perfectly straight gradient-based control trajectories is derived wherein the gradient on the quantum control landscape must be an eigenfunction of the Hessian. This relation is an indicator of landscape structure and may provide a means to identify physical conditions when control trajectories can achieve perfect linearity. The collective favorable landscape topology and structure provide a foundation to understand why optimal quantum control can be readily achieved.
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