Basis-Sensitive Quantum Typing via Realisability

2025-10-21

AI Summary

Existing quantum control λ-calculi rely on fixed-basis abstractions, limiting their ability to characterize program behavior under arbitrary (including entangled) bases and to reason about basis transformations. Method: We introduce λ_B—the first basis-agnostic quantum control λ-calculus grounded in real realizability semantics—featuring basis-sensitive abstraction and a let construct, together with a basis-dependent substitution mechanism for precise decomposition of value distributions. Crucially, we integrate unitary operators directly into the type system, enabling unified typing of unitary evolution and delayed measurement via typed unitary reduction rules. Contribution/Results: λ_B ensures type safety and faithfully encodes key quantum algorithms—including Deutsch’s algorithm and quantum teleportation—within a single, coherent type system. It provides fine-grained, formally verified modeling of entanglement, basis change, and measurement delay—core phenomena previously challenging to capture uniformly in quantum λ-calculi.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We present $λ_B$, a quantum-control $λ$-calculus that refines previous basis-sensitive systems by allowing abstractions to be expressed with respect to arbitrary -- possibly entangled -- bases. Each abstraction and let construct is annotated with a basis, and a new basis-dependent substitution governs the decomposition of value distributions. These extensions preserve the expressive power of earlier calculi while enabling finer reasoning about programs under basis changes. A realisability semantics connects the reduction system with the type system, yielding a direct characterisation of unitary operators and ensuring safety by construction. From this semantics we derive a validated family of typing rules, forming the foundation of a type-safe quantum programming language. We illustrate the expressive benefits of $λ_B$ through examples such as Deutsch's algorithm and quantum teleportation, where basis-aware typing captures classical determinism and deferred-measurement behaviour within a uniform framework.

Problem

• Develops a quantum-control calculus with basis-sensitive abstractions
• Establishes realisability semantics connecting reduction and type systems
• Enables finer reasoning about quantum programs under basis changes

Innovation

• Quantum calculus with basis-annotated abstractions
• Basis-dependent substitution for value distributions
• Realisability semantics connecting reduction and type systems