Quantum Control in the Unitary Sphere: Lambda-S1 and its Categorical Model

Abstract

In a recent paper, a realizability technique has been used to give a semantics of a quantum lambda calculus. Such a technique gives rise to an infinite number of valid typing rules, without giving preference to any subset of those. In this paper, we introduce a valid subset of typing rules, defining an expressive enough quantum calculus. Then, we propose a categorical semantics for it. Such a semantics consists of an adjunction between the category of semi-vector spaces of value distributions (that is, linear combinations of values in the lambda calculus), and the category of sets of value distributions.

Linear lambda-calculus is linear

“…Another issue is to restrict the logic further so that linear functions are unitary. We can either enforce unitarity, following the methods of [1,6,7], or let these unitarity constraints as properties of the program that must be proved for each program, rather than enforced by the type system.…”

A New Connective in Natural Deduction, and its Application to Quantum Computing

“…Indeed, to enforce the linearity of proofs of Q ⇒ Q, we can, for instance, restrict them to proofs of the form λx (App M x), for some proof M of B ⇒ Q. Then, to enforce unitarity, we can restrict the construction of M to matrices such that the columns M 0 and M 1 are orthogonal closed irreducible proofs [19]- [21]. Although such restrictions are strong, they preserve the quantum completeness of the calculus.…”