Diagrammatic Analysis for Parameterized Quantum Circuits

Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis.

In this work, we describe extensions of the ZX calculus for computing expectation values of parameterized quantum circuits,

which are important quantities in applications ranging from combinatorial optimization to quantum chemistry.

We provide several new ZX rules and generalizations for this setting.

In particular, we give formal rules for linear combinations of ZX diagrams,

where the relative scale factors of each diagram must be kept track of,

in contrast to most previously studied single-diagram cases where these coefficients can be ignored.

We demonstrate that the diagrammatic approach offers novel insights into algorithm structure and performance by considering

several ansätze from the literature including realizations of hardware-efficient ansätze and QAOA.

Moreover, we find that by using a diagrammatic representation the calculation of more sophisticated ansätze, like

deeper QAOA circuits, can become more intuitive and easier to calculate than by alternative means.

Finally, we outline how diagrammatic approaches may aid in the design of new and more effective quantum circuit ansätze.