This paper introduces the hybrid quantum language with general recursion $\mathtt{Hyrql}$, driven towards resource-analysis. By design, $\mathtt{Hyrql}$ does not require the specification of an initial set of quantum gates. Hence, it is well amenable towards a generic cost analysis, unlike languages that use different sets of quantum gates, which yield quantum circuits of distinct complexity. Regarding resource-analysis, we show how to relate the runtime of an expressive fragment of $\mathtt{Hyrql}$ programs with the size of the corresponding quantum circuits. We also manage to capture the class of functions computable in quantum polynomial time, which, by Yao's Theorem, corresponds to families of circuits of polynomial size. Consequently, this result paves the way for the use of termination and runtime-analysis techniques designed for classical programs to guarantee bounds on the size of quantum circuits.

翻译：本文介绍了一种面向资源分析的混合量子语言$\mathtt{Hyrql}$，该语言支持通用递归。$\mathtt{Hyrql}$在设计上无需指定初始量子门集合，因此非常适合进行通用成本分析，这与使用不同量子门集合（会产生不同复杂度量子电路）的语言形成鲜明对比。在资源分析方面，我们展示了如何将$\mathtt{Hyrql}$程序中一个表达性片段的运行时间与对应量子电路的规模相关联。我们还成功刻画了量子多项式时间可计算函数类，根据姚氏定理，这类函数对应于多项式规模电路族。因此，该结果为运用经典程序设计的终止性与运行时间分析技术来保证量子电路规模的上界开辟了道路。