We investigate the optimization of quantum control from a differential geometric perspective. In our approach, optimal control minimizes the cost associated with evolving a quantum state, with the cost quantified by the length of the trajectory on a relevant Riemannian manifold. We demonstrate the optimization protocol in systems with SU(2) and SU(1,1) dynamical symmetries, which encompass a broad range of physical systems. In these systems, the time evolution can be represented by trajectories on a three-dimensional manifold. Given the initial and final states, the minimum-cost quantum control corresponds to a geodesic on the manifold. When the trajectory between the initial and final states is specified, the minimum-cost control corresponds to a geodesic within a submanifold embedded in the three-dimensional space. This framework provides a geometric method for optimizing shortcuts to adiabatic driving.
Multi-mode interferometers for bosons in Gaussian states are important systems for quantum metrology with precision beyond the standard quantum limit and for bosonic quantum computing. However, there is a lack of theoretical foundation for generic
N-mode Gaussian interferometry. In this work, we study quantum metrology and quantum control in multi-mode bosonic systems with quadratic Hamiltonians, exploiting the fundamental Sp
(2N,R) symmetry of such interferometers. We show that the optimal quantum control to maximize sensitivity requires aligning squeezing and displacement in the same direction. We propose Sp
(2N,R) echo, a multi-mode generalization of the SU
(1,1) interferometry, to achieve the sensitivity of phase estimation set by the quantum Fisher information. In addition, we introduce a geometrical means for reversing many-body dynamics with Sp
(2N,R) dynamical symmetry, such as dynamics of the bosonic Kitaev chain. Our schemes are readily realizable in optical, atomic, and mechanical platforms.
苏(1， 1 ) 动态对称具有在在理论、适用的物理分析无界的量系统的基本重要性。在这份报纸，我们学习与苏一起与量系统联系的概括协调状态的控制(1， 1 ) 动态对称。在苏上基于一个假 Riemannian 度量标准(1， 1 ) 组，我们为最小化驾驶系统到需要的最后的状态的控制的领域 fluence 获得必要条件。进一步的分析证明候选人最佳的控制答案能被分类进正常、反常的 extremals。当控制 Hamiltonian 是非寓言的时，反常 extremals 能仅仅是经常的函数，当正常 extremals 能被 Weierstrass 椭圆形的函数根据控制 Hamiltonian 的 parabolicity 表示时。作为一种特殊情况，最大地挤压一个概括协调状态的最佳的控制解决方案是一个正弦曲线领域，它与在实验室被使用的一致。