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