发表机构
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明非通用变分量子电路在任意参考态下最大可达性与一般可达性等价，并给出可达性维度准则，数值验证了该准则对优化成败的决定性作用。
AI 中文摘要
采用与合适的态制备相匹配的特定问题非通用变分量子电路，已成为应对通用拟设训练困难的标准方法。然而，由于其非通用性，对其可达性和可训练性的诊断仍依赖于参考态。在本工作中，我们基于主轨道型定理，建立了任意参考态上最大可达性与一般可达性的等价性。此后，假设代价函数的全局最小值在某个可描述为光滑函数像的子集上取得，我们推导了在一般采样参考态下可达性概率非零的充分必要条件。此外，当解集被假定通过一个嵌入解流形的实解析映射实现时，我们证明若在单点处获得局部满射性，则在整个解流形上一般地获得局部满射性。作为实用设计规则，局部满射性的需求自动转化为必要的维度准则：可达性要求目标集的拓扑维度至少与一般轨道的余维度一样大。数值模拟表明，当维度障碍适用时，有限深度优化持续失败，而无障碍情况则显示出改进的收敛性。
英文摘要
Employing problem-specific non-universal Variational Quantum Circuits aligned with a suitable state preparation has become a standard approach to counter the difficulty in training universal ansätze. However, due to their non-universality, diagnosing their reachability and trainability remains reference-state-specific. In this work, we establish the equivalence of maximal reachability and generic reachability over arbitrary reference states as a consequence of the \emph{principal orbit-type theorem}. Thereafter, assuming that the global minimum of the cost function is achieved on a subset that can be described as the image of a smooth function, we derive necessary and sufficient conditions for non-zero probability of reachability under generically sampled reference states. Furthermore, when the solution set is assumed to be realised through a real analytic map that embeds a solution manifold, we show that local surjectivity is generically obtained on the entire solution manifold if it is attained at a single point. As a practical design rule, the need for local surjectivity automatically translates to a necessary dimensional criterion: reachability requires the target set's topological dimension to be at least as large as the co-dimension of the generic orbit. Numerical simulations show that finite-depth optimisation consistently fails when the dimensional obstruction applies, while unobstructed cases show improved convergence.
Comments13 pages, 3 figures, 3 tables
