Feynman Meets Turing: The Curse of Quantum Universality

Abstract

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.