Quantum superpositions and projective measurement in the lambda calculus

We propose an extension of simply typed lambda-calculus to handle some properties of quantum computing. The equiprobable quantum superposition is taken as a commutative pair and the quantum measurement as a non-deterministic projection over it. Destructive interferences are achieved by introducing an inverse symbol with respect to pairs. The no-cloning property is ensured by using a combination of syntactic linearity with linear logic. Indeed, the syntactic linearity is enough for unitary gates, while a function measuring its argument needs to enforce that the argument is used only once.