Towards a Computational Quantum Logic: An Overview of an Ongoing Research Program

Abstract: This invited paper presents an overview of an ongoing research program aimed at extending the Curry-Howard-Lambek correspondence to quantum computation. We explore two key frameworks that provide both logical and computational foundations for quantum programming languages. The first framework, the Lambda-$S$ calculus, extends the lambda calculus by incorporating quantum superposition, enforcing linearity, and ensuring unitarity, to model quantum control. Its categorical semantics establishes a structured connection between classical and quantum computation through an adjunction between Cartesian closed categiries and additive symmetric monoidal closed categories. The second framework, the $\mathcal L^{\mathbb C}$ calculus, introduces a proof language for intuitionistic linear logic augmented with sum and scalar operations. This enables the formal encoding of quantum superpositions and measurements, leading to a computational model grounded in categorical structures with biproducts. These approaches suggest a fundamental duality between quantum computation and linear logic, highlighting structural correspondences between logical proofs and quantum programs. We discuss ongoing developments, including extensions to polymorphism, categorical and realizability models, as well as the integration of the modality !, which further solidify the connection between logic and quantum programming languages.