费曼遇见图灵：量子通用性的诅咒
Feynman Meets Turing: The Curse of Quantum Universality
AI总结：
研究量子电路描述语言（QCDLs），证明语义通用的QCDL不存在可半判定的语义有意义描述集，即无可靠识别所有有效程序描述的编译器，揭示其与经典编程语言的本质区别及量子计算模型的基本限制。
AI中文摘要：
我们考虑量子电路描述语言（QCDLs）的形式模型，其中语义有意义的程序对应于可计算酉矩阵。我们表明，任何语义通用的QCDL——即任何能够描述所有可计算酉矩阵的QCDL，这些矩阵又构成了我们能在数字硬件上有意义地表示的矩阵集——都不可能有一个可半判定的语义有意义描述集。特别是，不存在这样一种语言能有一个可靠识别所有有效程序描述的编译器。这个结果与经典编程语言形成对比。在C或C++等语言中，编译本身可能涉及非终止计算，但语义有意义的程序集仍然是递归可枚举的，因为成功编译提供了有效性的证明。本质区别在于语义域的性质：经典语言描述部分递归函数，而QCDLs描述全酉算子。我们的分析确立了量子电路描述语言的一个基本限制，并突出了在形式语言理论层面经典和量子计算模型之间的结构差异。
英文摘要：
We consider a formal model of quantum circuit description languages (QCDLs) in which semantically meaningful programs correspond to computable unitary matrices. We show that any semantically universal QCDL -- that is, any QCDL able to describe all computable unitary matrices, which in turn form the set of matrices we can meaningfully represent on digital hardware -- cannot have a semi-decidable set of semantically meaningful descriptions. In particular, no such language admits a compiler that reliably recognizes all valid program descriptions. This result stands in contrast to classical programming languages. While compilation in languages such as C or C++ may itself involve non-terminating computations, the set of semantically meaningful programs remains recursively enumerable, since successful compilation provides a witness of validity. The essential difference lies in the nature of the semantic domains: classical languages describe partial recursive functions, whereas QCDLs describe total unitary operators. Our analysis establishes a fundamental limitation of quantum circuit description languages and highlights a structural distinction between classical and quantum models of computation at the level of formal language theory.
