Modern quantum programming languages integrate quantum resources and classical control. They must, on the one hand, be linearly typed to reflect the no-cloning property of quantum resources. On the other hand, high-level and practical languages should also support quantum circuits as first-class citizens, as well as families of circuits that are indexed by some classical parameters. Quantum programming languages thus need linear dependent type theory. This paper defines a general semantic structure for such a type theory via certain fibrations of monoidal categories. The categorical model of the quantum circuit description language Proto-Quipper-M by Rios and Selinger (2017) constitutes an example of such a fibration, which means that the language can readily be integrated with dependent types. We then devise both a general linear dependent type system and a dependently typed extension of Proto-Quipper-M, and provide them with operational semantics as well as a prototype implementation.

@article{paperbot1657,
  title = {Linear Dependent Type Theory for Quantum Programming Languages},
  author = {Peng Fu and Kohei Kishida and Peter Selinger},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 18, Issue 3},
  year = {2022},
  doi = {10.46298/lmcs-18(3:28)2022}
}