Diagrammatic Differentiation for Quantum Machine Learning

We introduce diagrammatic differentiation for tensor calculus by generalising the dual number construction from rigs to monoidal categories. Applying this ZX diagrams, we show how calculate diagrammatically gradient of a linear map with respect phase parameter. For diagrams parametrised quantum circuits, get well-known parameter-shift rule at basis many variational algorithms. then extend our method automatic differentation hybrid classical-quantum using bubbles encode arbitrary non-linear operators. Moreover, comes an open-source implementation in DisCoPy, Python library Diagrammatic gradients circuits can be simplified PyZX and executed on hardware via tket compiler. This opens door practical applications harnessing both structure string computational power machine learning.