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arXiv:2504.18441

Expectation-based Analysis of Higher-Order Quantum Programs

Martin Avanzini, Alejandro Díaz-Caro, Emmanuel Hainry F, Romain Péchoux

Published Apr 25, 2025• 0 citations
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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.

DOI
10.48550/arxiv.2504.18441

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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.
We introduce a new kind of expectation transformer for a mixed classical-quantum programming language. Our semantic approach relies on a new notion of a cost structure, which we introduce and which can be seen as a specialisation of the Kegelspitzen of Keimel and Plotkin. We show that our weakest precondition analysis is both sound and adequate with respect to the operational semantics of the language. Using the induced expectation transformer, we provide formal analysis methods for the expected cost analysis and expected value analysis of classical-quantum programs. We illustrate the usefulness of our techniques by computing the expected cost of several well-known quantum algorithms and protocols, such as coin tossing, repeat until success, entangled state preparation, and quantum walks.
Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive characterization of convex sets of maps of a given type is used to prove equivalence relations between different types. The axioms of the framework do not refer to the specific mathematical structure of quantum theory, and can therefore be exported in the context of any operational probabilistic theory.
In recent years, quantum computing has gained a substantial amount of momentum, and the capabilities of quantum devices are continually expanding and improving. Nevertheless, writing a quantum program from scratch remains tedious and error-prone work, showcasing the clear demand for automated tool support. We present Qet, a fully automated static program analysis tool that yields a precise expected cost analysis of mixed classical-quantum programs. Qet supports programs with advanced features like mid-circuit measurements and classical control flow. The methodology of our prototype implementation is based on a recently proposed quantum expectation transformer framework, generalising Dijkstra's predicate transformer and Hoare logic. The prototype implementation Qet is evaluated on a number of case studies taken from the literature and online references. Qet is able to fully automatically infer precise upper bounds on the expected costs that previously could only be derived by tedious manual calculations.
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We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify how classical probabilistic effects are induced by quantum measurements, and conversely, how classical (probabilistic) programs can influence the quantum dynamics. We also describe a sound and computationally adequate denotational semantics for the language. Classical probabilistic effects are interpreted using a recently-described commutative probabilistic monad on DCPO. Quantum effects and resources are interpreted in a category of von Neumann algebras that we show is enriched over (continuous) domains. This strong sense of enrichment allows us to develop novel semantic methods that we use to interpret the relationship between the quantum and classical probabilistic effects. By doing so we provide a very detailed denotational analysis that relates domain-theoretic models of classical probabilistic programming to models of quantum programming.
Quantum weakest preconditions are a fundamental tool for program verification of quantum programs. Many variations have been reported in the literature. We revisit quantum weakest preconditions from the perspective of expected runtime analysis of quantum programs and introduce a novel pre-expectation framework that enables to reason about the preconditions of quantum programs without the need of an upper bound. This is particularly interesting for quantum programs involving reward statements. The overall goal is to analyze runtime behavior even in the case of programs with potentially infinite expected runtime. This paper presents several ways to do so, e.g., a program transformation such that the expected runtime of a quantum program can be expressed using the weakest pre-expectation calculus with rewards.