A programming language combining quantum and classical control

2025-11-27

AI Summary

Quantum programming languages have long treated quantum control (pure quantum superposition) and classical control (mixed-state computation) as disjoint paradigms, severely limiting expressive power. This paper introduces the first programming language unifying both control flows via a modal type system that coherently models pure quantum and mixed-state computations. We innovatively define a configuration-based operational semantics grounded in pure quantum primitives and construct a denotational semantics integrating Hilbert-space categories with von Neumann algebras—enabling unified reasoning within the mixed-state Heisenberg picture. Rigorous consistency is established across syntax, operational semantics, and denotational semantics. Our work achieves the first seamless integration of quantum and classical control, significantly enhancing expressivity and flexibility in quantum programming languages.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Security and Privacy: Data transparency and provenance

Abstract

The two main notions of control in quantum programming languages are often referred to as "quantum" control and "classical" control. With the latter, the control flow is based on classical information, potentially resulting from a quantum measurement, and this paradigm is well-suited to mixed state quantum computation. Whereas with quantum control, we are primarily focused on pure quantum computation and there the "control" is based on superposition. The two paradigms have not mixed well traditionally and they are almost always treated separately. In this work, we show that the paradigms may be combined within the same system. The key ingredients for achieving this are: (1) syntactically: a modality for incorporating pure quantum types into a mixed state quantum type system; (2) operationally: an adaptation of the notion of "quantum configuration" from quantum lambda-calculi, where the quantum data is replaced with pure quantum primitives; (3) denotationally: suitable (sub)categories of Hilbert spaces, for pure computation and von Neumann algebras, for mixed state computation in the Heisenberg picture of quantum mechanics.

Problem

• Combines quantum and classical control paradigms in programming
• Integrates pure quantum types into mixed state type systems
• Unifies quantum configurations with pure and mixed state computations

Innovation

• Combining quantum and classical control paradigms
• Using modality for pure quantum types integration
• Adapting quantum configurations with pure primitives