Addition and Differentiation of ZX-diagrams

TL;DR

This paper introduces methods for adding and differentiating ZX-diagrams, enhancing the calculus's expressiveness and enabling applications like deriving diagrams for quantum Hamiltonians.

Contribution

It provides the first formal definitions for addition and differentiation of ZX-diagrams, expanding the calculus's capabilities for quantum reasoning.

Findings

Defined inductive addition of ZX-diagrams using controlled diagrams

Developed inductive and non-inductive differentiation techniques for ZX-diagrams

Applied differentiation to derive a diagram for an Ising Hamiltonian

Abstract

The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.