Recursive techniques have recently been introduced into quantum programming so that a variety of large quantum circuits and algorithms can be elegantly and compactly programmed. In this paper, we present a proof system for formal verification of the correctness of recursively defined quantum circuits. The soundness and (relative) completeness of the proof system are established. To demonstrate its effectiveness, we present a series of application examples, including formal verification of multi-qubit controlled gates, a quantum circuit for generating multi-qubit GHZ (Greenberger-Horne-Zeilinger) states, and more sophisticated quantum algorithms with recursive structures such as the quantum Fourier transform, quantum state preparation, and quantum random access memories (QRAMs).

@article{paperbot3717,
  title = {Verification of Recursively Defined Quantum Circuits},
  author = {Mingsheng Ying and Zhicheng Zhang},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {10},
  number = {PLDI},
  year = {2026},
  doi = {10.1145/3808273}
}