QWIRE Practice: Formal Verification of Quantum Circuits in Coq

Abstract: We describe an embedding of the QWIRE quantum circuit language in the Coq proof assistant. This allows programmers to write quantum circuits using high-level abstractions and to prove properties of those circuits using Coq's theorem proving features. The implementation uses higher-order abstract syntax to represent variable binding and provides a type-checking algorithm for linear wire types, ensuring that quantum circuits are well-formed. We formalize a denotational semantics that interprets QWIRE circuits as superoperators on density matrices, and prove the correctness of some simple quantum programs.