AI 中文总结
研究任意量子态矢量优化问题，提出霍普夫假设这一二叉树电路方法。该方法能提供坐标、距离信息等，用于态制备与逆映射。在确定性实态基准测试中表现良好，将通用态制备转变为可导航的任意纯态优化框架。
AI 中文摘要
优化任意量子态矢量就像在希尔伯特空间中的单位球面上导航：除了目标可达性，优化还需要坐标、局部距离信息和可测量方向。我们引入了霍普夫假设，这是一种用于任意归一化实部和复部态矢量的二叉树电路。内角控制子树之间的概率，叶相位携带复自由度，同一棵树给出态制备以及从振幅到物理角度的显式逆映射。这些结构共同作用，在搜索中充当指南针：逆映射给出坐标，对角诱导度量给出局部距离信息，非零坐标切线成为可制备的归一化态。对于哈密顿量目标以及具有相同局部跃迁矩形式的目标，每个梯度分量是一个已知比例因子乘以当前态与切线态之间的跃迁矩。分支态构造通过期望值测量来表达这些矩，而树则按幅度层和叶索引相位族来组织编译后的梯度设置。 因此，不同编译梯度访问电路族的数量仅随希尔伯特空间维度对数增长，而达到选定精度所需的测量预算仍然是单独的统计成本。在具有精确成本、精确梯度和已知全局最优的确定性实态基准测试中，度量感知霍普夫优化器达到了数值精度中位数差距，在较小的变分量子本征求解器平均最终差距中具有最明显的基线增益，并且在数值精度上计量启发式迹线的集中度更高。霍普夫假设将通用态制备转变为用于任意纯态优化的可导航框架。
英文摘要
Optimizing arbitrary quantum state vectors is like navigating the unit sphere in Hilbert space: beyond target reachability, optimization asks for coordinates, local distance information, and measurable directions. We introduce the Hopf ansatz, a binary-tree circuit for arbitrary normalized real and complex state vectors. Internal angles steer probability between subtrees, leaf phases carry the complex degrees of freedom, and the same tree gives state preparation and an explicit inverse map from amplitudes to physical angles. Together these structures act as a compass for the search: the inverse map gives coordinates, the diagonal induced metric gives local distance information, and nonzero coordinate tangents become preparable normalized states. For Hamiltonian objectives, and for objectives with the same local transition-moment form, each gradient component is a known scale factor times a transition moment between the current state and a tangent state. A branch-state construction expresses these moments through expectation-value measurements, while the tree organizes the compiled gradient settings by magnitude layer and leaf-indexed phase family. Thus the number of distinct compiled gradient-access circuit families grows only logarithmically with Hilbert-space dimension, while the measurement budget required for a chosen precision remains a separate statistical cost. In deterministic real-state benchmarks with exact costs, exact gradients, and known global optima, metric-aware Hopf optimizers reach numerical-precision median gaps, with the clearest baseline gains in smaller VQE mean final gaps and stronger concentration of metrology-inspired traces at numerical precision. The Hopf ansatz turns universal state preparation into a navigable framework for arbitrary pure-state optimization.
