ZX Graphical Calculus for Continuous-Variable Quantum Processes

Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators in the Heisenberg picture, and the features of CV space has yet to be thoroughly investigated in an intuitive manner. One key ingredient for further exploration of CV quantum computing is the construction of a computational model that brings visual intuition and new tools for analysis. In this paper, we delve into a graphical computational model, inspired by a similar model for qubit-based systems called the ZX calculus, that enables the representation of arbitrary CV quantum process as a simple directed graph. We demonstrate the utility of our model as a graphical tool to comprehend CV processes intuitively by showing how equivalences between two distinct quantum processes can be proven as a sequence of diagrammatic transformations in certain cases. We also examine possible applications of our model, such as measurement-based quantum computing, characterization of Gaussian and non-Gaussian processes, and circuit optimization.