This paper introduces QWire, a language for defining quantum circuits and an interface for manipulating them inside of an arbitrary classical host language. QWire is designed to be minimal—it contains only a few primitives—and sound with respect to the physical properties entailed by quantum mechanics. At the same time, QWire is expressive and highly modular due to its relationship with the host language, which mirrors the QRAM model of computation that places a quantum computer (controlled by circuits) alongside a classical computer (controlled by the host language).

We present QWire along with its type system and operational semantics, which we prove is safe and strongly normalizing whenever the host language is. We give circuits a denotational semantics in terms of density matrices. Throughout, we investigate examples that demonstrate the expressive power of QWire, including extensions to the host language that (1) expose a general analysis framework for circuits, and (2) provide dependent types.