量子包围控制景观的运动学“近临界”集的体积分数
Volume Fractions of the Kinematic "Near-Critical" Sets of the Quantum Ensemble Control Landscape
摘要：
导出了运动学量子系综控制景观J（U）= Tr（UρU' O）的临界集$C^{P}\simeq\mathrm{U}(\mathbf{n})P\mathrm{U}(\mathbf{m})$附近的子集$C_ε^{P} = \{U : ||grad J(U)|\leq ε\}\subset\mathrm{U}(N)$的体积分数的估计，其中$U$表示幺正时间演化算子，ρ是系综的初始密度矩阵，O是可观测算子.该估计基于酉群的希尔伯特-施密特几何和$||grad J(U)||^2$的一阶近似。这些近临界体积上的上限被证明和支持的数值模拟，导致渐近分析的尺寸$N$的量子系统的上升，其中这些“近临界”集的体积分数减少到零，作为$N$增加。这一结果有助于解释$J$的许多鞍对梯度流明显缺乏影响。
英文摘要：
An estimate is derived for the volume fraction of a subset $C_ε^{P} = \{U : ||grad J(U)|\leq ε\}\subset\mathrm{U}(N)$ in the neighborhood of the critical set $C^{P}\simeq\mathrm{U}(\mathbf{n})P\mathrm{U}(\mathbf{m})$ of the kinematic quantum ensemble control landscape J(U) = Tr(UρU' O), where $U$ represents the unitary time evolution operator, ρ is the initial density matrix of the ensemble, and O is an observable operator. This estimate is based on the Hilbert-Schmidt geometry for the unitary group and a first-order approximation of $||grad J(U)||^2$. An upper bound on these near-critical volumes is conjectured and supported by numerical simulation, leading to an asymptotic analysis as the dimension $N$ of the quantum system rises in which the volume fractions of these "near-critical" sets decrease to zero as $N$ increases. This result helps explain the apparent lack of influence exerted by the many saddles of $J$ over the gradient flow.
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