Quantum Orchestras: a Concrete Semantics for Recursive Hybrid Programs

2026-07-10

AI Summary

This work addresses the lack of formal semantics for hybrid quantum programs that integrate classical control flow with quantum effects, particularly in supporting mid-circuit measurements, recursion, and non-terminating behaviors. To this end, the paper proposes a denotational semantic framework grounded in the theory of quantum instruments. By formally embedding quantum instruments into the category of directed-complete partial orders (DCPOs) and constructing a quantum orchestra monad, the framework uniformly captures quantum non-commutativity alongside classical state and probabilistic effects. This approach yields the first semantic model capable of naturally accommodating recursion, mid-circuit measurement, and divergence, while generalizing both the classical state monad and the probabilistic powerdomain monad. The resulting foundation provides a rigorous and expressive semantic basis for reasoning about hybrid quantum programs.

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.

Problem

• quantum programming languages
• hybrid quantum computations
• mid-circuit measurements
• non-termination
• denotational semantics

Innovation

• quantum orchestra monad
• quantum instruments
• denotational semantics
• hybrid quantum programs
• DCPO