AI 中文总结
研究递归混合量子程序，引入分级图序列语义，构建语义评估并证明其兼容性，通过截断执行序列恢复最小不动点表示，用\(q^n\)加权得阿贝尔正则化语义，还涉及线性反馈扇区及弗雷德霍姆反馈行列式。
AI 中文摘要
我们为在量子管弦乐队单子中解释的递归混合量子程序引入了一种分级图序列语义。有限终止执行由有向路径表示，其边携带正规完全正子单位映射，终端顶点携带经典结果。路径连接定义了一个分级执行范畴，而延续嫁接对依赖结果的顺序组合进行建模。我们构建了从可允许执行图序列到量子管弦乐队的语义评估，并证明它与通道组合和克莱斯利组合都兼容。对于有限递归程序，执行序列在度数\(n\)处的截断与相关斯科特连续递归泛函的第\(n\)个克莱因近似一致。因此，完整图序列的评估恢复了普通最小不动点表示。用\(q^n\)加权度数为\(n\)的图，其中\(0<q<1\)，产生一种阿贝尔正则化语义，其当\(q\to 1^{-}\)时的斯科特极限是未正则化的递归表示。等效地，参数化\(q = e^{-t}\)指数抑制长执行，并在\(t\to {0}^{+}\)时重建表示。在补充线性反馈扇区中，重复递归由执行预解式\((I - qST)^{-1}\)表示。我们将\(I - qST\)与图子空间的代数交比识别。在希尔伯特 - 施密特假设下，相关返回算子是迹类的，并定义了弗雷德霍姆反馈行列式\(\operatorname{det}_{F}(I - qST)\)，其零点检测奇异反馈配置，其对数展开记录闭环遍历。
英文摘要
We introduce a graded graph-series semantics for recursive hybrid quantum programs interpreted in the quantum orchestra monad. Finite terminating executions are represented by directed paths whose edges carry normal completely positive subunital maps and whose terminal vertices carry classical results. Path concatenation defines a graded execution category, while continuation grafting models outcome-dependent sequential composition. We construct a semantic evaluation from admissible execution-graph series to quantum orchestras and prove that it is compatible with both channel composition and Kleisli composition. For finitary recursive programs, the truncation of the execution series at degree $n$ is shown to coincide with the $n$-th Kleene approximant of the associated Scott-continuous recursion functional. Consequently, evaluation of the complete graph series recovers the ordinary least-fixed-point denotation. Weighting a graph of degree $n$ by $q^n$, with $0<q<1$, yields an Abel-regularised semantics whose Scott limit as $q\to 1^{-}$ is the unregularised recursive denotation. Equivalently, the parametrisation $q=e^{-t}$ exponentially suppresses long executions and reconstructs the denotation as $t\to 0^{+}$. In a supplementary linear feedback sector, repeated recursion is represented by the execution resolvent $(I-qST)^{-1}$. We identify $I-qST$ with an algebraic cross-ratio of graph subspaces. Under Hilbert--Schmidt assumptions, the associated return operator is trace class and defines the Fredholm feedback determinant \( \operatorname{det}_{F}(I-qST), \) whose zeros detect singular feedback configurations and whose logarithmic expansion records closed loop traversals.
