A Quantum-Control Lambda-Calculus with Multiple Measurement Bases

2025-06-19

AI Summary

This paper addresses the challenge of flexibly supporting measurements in arbitrary bases within quantum programming languages. We propose the first type system that natively embeds arbitrary measurement bases into types. To this end, we introduce Λ-SX, a typed quantum lambda calculus wherein types explicitly track basis-dependent duplicability—thereby unifying the resource behavior and compositional semantics across different measurement bases. Our design overcomes the traditional single-basis restriction, enabling compositional reasoning and safe control for cross-basis measurements. By integrating quantum operational semantics with a subtyping system, we establish a fully formal framework and rigorously prove key metatheoretic properties: strong normalization, progress, and type preservation. This work provides a type-theoretic foundation for multi-basis quantum programming that is expressive, safe, and extensible.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We introduce Lambda-SX, a typed quantum lambda-calculus that supports multiple measurement bases. By tracking duplicability relative to arbitrary bases within the type system, Lambda-SX enables more flexible control and compositional reasoning about measurements. We formalise its syntax, typing rules, subtyping, and operational semantics, and establish its key meta-theoretical properties. This proof-of-concept shows that support for multiple bases can be coherently integrated into the type discipline of quantum programming languages.

Problem

• Introducing Lambda-SX for quantum control with multiple measurement bases
• Enabling flexible measurement control via typed lambda-calculus
• Formalizing syntax and semantics for multi-basis quantum programming

Innovation

• Typed quantum lambda-calculus with multiple bases
• Duplicability tracking via type system
• Formalized syntax and semantics for quantum control