Hybrid Quantum-Classical Machine Learning with String Diagrams

Central to near-term quantum machine learning is the use of hybrid quantum-classical algorithms. This paper develops a formal framework for describing these algorithms in terms of string diagrams: a key step towards integrating these hybrid algorithms into existing work using string diagrams for machine learning and differentiable programming. A notable feature of our string diagrams is the use of functor boxes, which correspond to a quantum-classical interfaces. The functor used is a lax monoidal functor embedding the quantum systems into classical, and the lax monoidality imposes restrictions on the string diagrams when extracting classical data from quantum systems via measurement. In this way, our framework provides initial steps toward a denotational semantics for hybrid quantum machine learning algorithms that captures important features of quantum-classical interactions.