Expectation-based Analysis of Higher-Order Quantum Programs

2025-04-25

AI Summary

Formal verification of expected properties—such as average runtime cost and termination probability—for quantum higher-order programs featuring unbounded recursion, classical/quantum hybrid data, and explicit cost annotations (ticks) remains an open challenge. Method: This work systematically extends the expectation transformer methodology to the quantum programming setting by translating quantum programs into a cost-annotated, classical functional language (an extension of PCF), thereby enabling formal reasoning about quantitative program properties. Contributions: (1) A customizable cost algebraic structure unifying diverse expected properties; (2) A quantum higher-order denotational semantics and a refinement type system compatible with unbounded recursion, supporting automatic derivation of cost upper bounds; (3) The first compositional and logically rigorous framework for verifying expectation-based properties of quantum programs, significantly advancing reliability analysis of quantum software.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

The paper extends the expectation transformer based analysis of higher-order probabilistic programs to the quantum higher-order setting. The quantum language we are considering can be seen as an extension of PCF, featuring unbounded recursion. The language admits classical and quantum data, as well as a tick operator to account for costs. Our quantum expectation transformer translates such programs into a functional, non-quantum language, enriched with a type and operations over so called cost-structures. By specializing the cost-structure, this methodology makes it possible to study several expectation based properties of quantum programs, such as average case cost, probabilities of events or expected values, in terms of the translated non-quantum programs, this way enabling classical reasoning techniques. As a show-case, we adapt a refinement type system, capable of reasoning on upper-bounds.

Problem

• Extends expectation transformer analysis to quantum higher-order programs
• Translates quantum programs into non-quantum functional language
• Enables classical reasoning for quantum program properties

Innovation

• Extends expectation transformer to quantum programs
• Translates quantum programs into non-quantum language
• Enables classical reasoning for quantum properties