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arXiv:2607.09605

Quantum Orchestras: a Concrete Semantics for Recursive Hybrid Programs

Alex Rice, Dominik Leichtle, Kim Worrall, Robert I. Booth

cs.PLcs.LOquant-phSubmitted Jul 10, 2026• Announced Jul 13, 2026• 0 citations
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Abstract

Many production quantum programming languages represent hybrid quantum computations by extending a classical base language with a quantum effect, where qubits are addressed by reference, and quantum operations are understood to mutate some external quantum state. However, the semantics of this view of quantum computation remains underdeveloped, especially when the language allows mid-circuit measurements and non-termination. In this work, we provide a general method for building denotational semantics for such languages, by defining the quantum orchestra monad, which precisely captures this style of quantum effect. The monad has a concrete presentation, being based on the formalism of quantum instruments, a common tool in quantum information theory for capturing the action of a quantum process along with its classical outcomes. It acts on the category DCPO, and so enables the interpretation of divergent hybrid programs. The quantum orchestra monad serves as a natural extension of both the classical state monad and the probabilistic powerdomain monad. We investigate some of the subtleties present when trying to naïvely extend these definitions to the quantum non-commutative case.

Comments
25 pages + 23 appendix

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Many production quantum programming languages represent hybrid quantum computations by extending a classical base language with a quantum effect, where qubits are addressed by reference, and quantum operations are understood to mutate some external quantum state. However, the semantics of this view of quantum computation remains underdeveloped, especially when the language allows mid-circuit measurements and non-termination.
In this work, we provide a general method for building denotational semantics for such languages, by defining the quantum orchestra monad, which precisely captures this style of quantum effect. The monad has a concrete presentation, being based on the formalism of quantum instruments, a common tool in quantum information theory for capturing the action of a quantum process along with its classical outcomes. It acts on the category DCPO, and so enables the interpretation of divergent hybrid programs.
The quantum orchestra monad serves as a natural extension of both the classical state monad and the probabilistic powerdomain monad. We investigate some of the subtleties present when trying to naïvely extend these definitions to the quantum non-commutative case.
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.
We introduce a graded graph-series semantics for recursive hybrid quantum programs interpreted in the quantum orchestra monad. Finite terminating executions are represented by directed paths whose edges carry normal completely positive subunital maps and whose terminal vertices carry classical results. Path concatenation defines a graded execution category, while continuation grafting models outcome-dependent sequential composition. We construct a semantic evaluation from admissible execution-graph series to quantum orchestras and prove that it is compatible with both channel composition and Kleisli composition. For finitary recursive programs, the truncation of the execution series at degree nnn is shown to coincide with the nnn-th Kleene approximant of the associated Scott-continuous recursion functional. Consequently, evaluation of the complete graph series recovers the ordinary least-fixed-point denotation. Weighting a graph of degree nnn by qnq^nqn, with $0
This thesis revolves around an area of computer science called "semantics". We work with operational semantics, equational theories, and denotational semantics. The first contribution of this thesis is a study of the commutativity of effects through the prism of monads. Monads are the generalisation of algebraic structures such as monoids, which have a notion of centre: the centre of a monoid is made of elements which commute with all others. We provide the necessary and sufficient conditions for a monad to have a centre. We also detail the semantics of a language with effects that carry information on which effects are central. Moreover, we provide a strong link between its equational theories and its denotational semantics. The second focus of the thesis is quantum computing, seen as a reversible effect. Physically permissible quantum operations are all reversible, except measurement; however, measurement can be deferred at the end of the computation, allowing us to focus on the reversible part first. We define a simply-typed reversible programming language performing quantum operations called "unitaries". A denotational semantics and an equational theory adapted to the language are presented, and we prove that the former is complete. Furthermore, we study recursion in reversible programming, providing adequate operational and denotational semantics to a Turing-complete, reversible, functional programming language. The denotational semantics uses the dcpo enrichment of rig join inverse categories. This mathematical account of higher-order reasoning on reversible computing does not directly generalise to its quantum counterpart. In the conclusion, we detail the limitations and possible future for higher-order quantum control through guarded recursion.
Monads are a ubiquitous structure in functional programming used for modelling computational effects. For example, the state monad models the effect of a computation interacting with a memory system. Here we introduce the quantum instrument monad IA\mathcal{I}_\mathcal{A}IA​, which models the effect of a computation interacting with a quantum system with algebra of observables A\mathcal{A}A. It can be thought of as a noncommutative generalization of the state monad. We construct this quantum instrument monad in two versions: a finitary version on the category of sets and a measure-theoretic version on the category of measurable spaces (the latter under the assumption that A\mathcal{A}A is a type I von Neumann algebra with separable predual). Both versions are strong monads. The construction of the measure-theoretic version is based on a new notion of integral of a quantum-operation-valued function against a state-valued measure.
With the approaching end of Moore’s Law, where will not be possible to improve the capacity of silicon based processors, the quantum computing appear to be a good choice to provide a new era of computation. Quantum computing can be understood as the art of transform information encoded in the state of a quantum physical system. This encoding is through the quantum bits (qubits), which can be on superposition or entangled states, enabling to explore the property called quantum parallelism. In this work is discussed the creation of a quantum programming language implementing the object-oriented paradigm (OO), allowing manipulation of classes and objects, where the quantum effects are handled through a monadic approach, extending the Featherweight Java (FJ) proposal. This language is formally defined through the operational semantics, which allow the implementation in any language that provides closures. That language formalization enables us to create an interpreter, implementing the steps of lexical, syntactic and semantic analysis, focusing in the type system to embedded quantum computing concepts in a classical language. Several examples are provided in the text, showing ways to handle the monadic layer in order to perform transformations in quantum information.
Based on the similarity between the categorial derivation of classical programs from their specification and the category theory approach to quantum physics, this dissertation aims at extending the laws of classical program algebra to quantum programming. In this context, the principles of the algebra of classical programs are applied to quantum programming, in order to verify the feasibility of creating correct-by-construction quantum circuits that can run on quantum devices available in the IBM Q Experience. The reversibility restrictions of quantum circuits are ensured by minimal complements. Moreover, measurements are postponed to the end of recursive computations called “quantamorphisms” to avoid the collapse of quantum states. Quantamorphisms are classical catamorphisms extended to ensure quantum reversibility. The derived quantamorphisms implement quantum cycles (vulg. for-loops) and quantum folds on lists. By Kleisli correspondence, quantamorphisms can be written as monadic functional programs with quantum parameters. This enables the use of Haskell, a monadic functional programming language, to perform the experimental work. The examples of the calculated quantum programs are simulated in Haskell, Quipper and QISKit and run on the quantum computers of the IBM Q Experience. The main conclusions of this work are that, while all the simulations produced correspond to the predicted results, running these programs on real quantum devices results in a significant amount of errors. As quantum devices are constantly evolving, it is likely that in the near future these devices will increase their reliability, allowing programs to run more accurately. The extension of the quantamorphism concept to more general input structures, such as finite trees, remains a challenge that is left for future work. Also relevant will be the study of conditional quantum control without measurements, which will extend the scope of quantamorphisms as quantum circuit specifications.