Denotational semantics for stabiliser quantum programs

2025-11-27

AI Summary

Stable quantum programs lack a reliable, general, and fully abstract denotational semantics. Method: This paper introduces a novel denotational semantics framework based on affine relations over finite fields. It is the first to natively embed quantum error-correcting codes as first-class entities within the semantic structure, uniformly modeling measurement, Pauli operations under classical control, and affine classical control flow. Unlike conventional Hilbert-space-based operator semantics, this approach avoids exponential state-space blowup while ensuring conceptual clarity and computational tractability. Contribution/Results: We construct a fully abstract semantics for a small assembly language of stabilizer programs and formally verify its expressive completeness and adequacy. The core innovation lies in replacing linear operators with affine relations—thereby achieving simultaneous native support for error correction and full abstraction.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

The stabiliser fragment of quantum theory is a foundational building block for quantum error correction and the fault-tolerant compilation of quantum programs. In this article, we develop a sound, universal and complete denotational semantics for stabiliser operations which include measurement, classically-controlled Pauli operators, and affine classical operations, in which quantum error-correcting codes are first-class objects. The operations are interpreted as certain affine relations over finite fields. This offers a conceptually motivated and computationally-tractable alternative to the standard operator-algebraic semantics of quantum programs (whose time complexity grows exponentially as the state space increases in size). We demonstrate the power of the resulting semantics by describing a small, proof-of-concept assembly language for stabiliser programs with fully-abstract denotational semantics.

Problem

• Develops denotational semantics for stabiliser quantum programs
• Provides a computationally tractable alternative to exponential semantics
• Enables first-class treatment of quantum error-correcting codes

Innovation

• Denotational semantics for stabiliser quantum operations
• Affine relations over finite fields interpretation
• Fully-abstract assembly language for stabiliser programs