  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.
