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

2025-04-10

AI Summary

This work establishes a logical and type-theoretic foundation for quantum computation by extending the Curry–Howard–Lambek correspondence to the quantum domain. To address the challenge of formally capturing quantum phenomena—such as superposition, measurement, and unitary evolution—two novel formal calculi are introduced: the Lambda-S calculus, which natively supports quantum control constructs, and the L^C calculus, grounded in intuitionistic linear logic. Crucially, this work establishes, for the first time at the structural level, a duality between quantum control and linear logic. Using adjoint functors, additive symmetric monoidal closed categories, and biproduct semantics, it unifies classical and quantum computation within a single categorical framework. The resulting theory yields a rigorous correspondence between quantum programs and logical proofs, enabling type-safe modeling of superposition encoding and measurement. This constitutes the first type theory for quantum programming that is both semantically sound and logically complete.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Extending Curry-Howard-Lambek to quantum computation
• Modeling quantum control via Lambda-$S calculus
• Encoding quantum superpositions with $mathcal{L}^{mathbb{C}}$ calculus

Innovation

• Lambda-$S$ calculus extends lambda calculus for quantum control
• $mathcal{L}^{mathbb{C}}$ calculus encodes quantum superpositions and measurements
• Duality between quantum computation and linear logic