Addressing problems beyond classical computing limits is sparking an increasing interest in Quantum Computing. However, despite their adequacy to address specific problems, quantum algorithms cover a limited subset of the functionality required in real-world computing systems. Additionally, they require expensive specialized hardware. To overcome this issue, hybrid (quantum-classical) software systems are emerging as a promising way to integrate both computing paradigms by applying the principles of Service-Oriented Architectures (SOA). Still, the design and deployment of hybrid service-based systems faces unique challenges like the idiosyncrasies and constraints of NISQ computers (e.g., algorithms that can only run in specific machines, disparate quality attribute metrics), and the management of structural and behavioural properties of service-based applications. From the SOA perspective, architectural decisions need to be made by performing a trade-off analysis and providing quantitative guarantees of system configurations under prescribed levels of uncertainty. In this paper, a method to explore the design space of quantum-classical applications is provided by a formalization of an architectural style of hybrid applications. The obtained results demonstrate that the proposed method successfully identifies decision boundaries. It enables the dynamic selection of the most suitable hybrid or classical configuration based on the user's QoS criteria.

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