发表机构
Tel-Hai University（泰尔海大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文重新审视量子控制目标，提出量子球面优化基准，证明其产率缺口可表示为 sin² 项线性组合，从而导出白盒梯度下降和黑盒进化策略的线性收敛速率，并验证了噪声鲁棒性。
AI 中文摘要
我们重新审视一个既定的量子控制目标，刻画其精确的局部度量几何，并将其重新定位为搜索与优化的严格基准。该景观的无陷阱拓扑结构在二十年前就已确立，保证了无障碍的最优路径，但全局拓扑本身并不决定收敛速度。尽管所谓的量子球面由控制场的四次方定义，我们证明其产率缺口可以精确表示为 sin² 项的非负线性组合。仅凭这一恒等式，我们推导出一个比率 4/π² 的双侧二次夹逼，作用于一个经过认证的最坏情况四分之一波区域，证明最优峰值在其仿射规范下非退化且良态。对于白盒优化，在一个减半（八分之一波）区域上，目标函数满足带显式常数的 Polyak-Łojasiewicz 不等式，赋予梯度下降线性速率 1 - 16/(π⁴κ_r)，其中 κ_r 是相关 Hessian 块的条件数。对于黑盒优化，我们证明精英单亲进化策略从认证的子水平集几乎必然地以线性速率收敛，且在期望命中时间内收敛，尽管景观具有连续规范对称性和周期最优点的晶格；当景观无陷阱时，认证水平是显式的，且在最优值附近，该策略可证明仅看到峰值的二次切面。系统性数值模拟证实了理论。最后，我们证明加性输入（执行器）噪声通过 Debye-Waller 因子重新加权表示中的每一项，而不破坏景观结构，并概述了开放挑战：输出（信号）噪声、远场平台逃逸、角同步认证，以及另一个物理可扩展的基准（含玻璃的 SHG）。
英文摘要
We revisit an established Quantum Control objective, characterize its exact local metric geometry, and reposition it as a rigorous benchmark for Search and Optimization. The trap-free topology of the landscape, established two decades ago, guarantees unhindered optimal pathways, but global topology alone does not govern convergence speed. Although the so-called Quantum Sphere is defined by the fourth power of the control field, we show that its yield gap admits an exact representation as a nonnegative linear combination of $\sin^2$ terms. From this single identity, we derive a two-sided quadratic sandwich with ratio $4/π^2$ on a certified, worst-case quarter-wave region, proving the optimal peak nondegenerate modulo its affine gauge and well-conditioned. For white-box optimization, on a halved (eighth-wave) region, the objective satisfies a Polyak-Łojasiewicz inequality with an explicit constant, granting Gradient Descent a linear rate of $1 - 16/(π^4κ_r)$, where $κ_r$ is the condition number of the relevant Hessian block. For black-box optimization, we prove that the elitist single-parent Evolution Strategy converges linearly, almost surely and in expected hitting time, from a certified sublevel set, despite the landscape's continuous gauge symmetry and its lattice of periodic optima; the certified level is explicit whenever the landscape is trap-free, and near the optimum the strategy provably sees only the quadratic tangent of the peak. Systematic numerical simulations corroborate the theory. Finally, we show that additive input (actuator) noise reweights each term of the representation by a Debye-Waller factor without disrupting the landscape's structure, and outline open challenges: output (signal) noise, far-field plateau escape, angular-synchronization certificates, and another physically scalable benchmark (SHG with Glass).
