In my master thesis I formally introduce sums of ZX-diagrams and propose a simplification algorithm for a specific type of ZX-diagram.
It can be found here
Abstract
ZX-calculus is a modern approach to quantum computing, that has also been applied to reasoning about solutions of combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) devices. Arising in applications for the Quantum Approximate Optimization Algorithm (QAOA) is the problem of simplifying symbolic ZX-diagrams to analytical expressions. We try to formalize the notion of previously proposed linear combinations of ZX-diagrams and harness their power towards a simplification algorithm. To our knowledge prior research does not formally define sums of arbitrary diagrams. Further no simplification algorithm known to us is capable of simplifying symbolic ZX-diagrams to an analytical expression. This twofold gap is tackled first by means of assessing category theoretic properties of the category ZX and then by developing a simplification algorithm using these properties. We formalize the notion of linear combinations using the structure of a category enriched over commutative monoids. As a first step towards analysing arbitrary QAOA circuits, we present an algorithm for simplifying symbolic ZX-diagrams that come up when applying QAOA to the Maximum Cut Problem. In summary we formally define linear combinations of ZX-diagrams and show the practical impact of this definition by proposing an algorithm that goes a step towards analysing the performance of QAOA.