2020/06/30 by Niel de Beaudrap
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.4204/eptcs.340.2
published as EPTCS 340, 2021, pp. 13-45 · In Proceedings QPL 2020, arXiv:2109.01534
arxiv created 2021/09/06 · arxiv updated 2021/09/07
The ZX calculus is a mathematical tool to represent and analyse quantum operations by manipulating diagrams which in effect represent tensor networks. Two families of nodes of these networks are ones which commute with either Z rotations or X rotations, usually called "green nodes" and "red nodes" respectively. The original formulation of the ZX calculus was motivated in part by properties of the algebras formed by the green and red nodes: notably, that they form a bialgebra -- but only up to scalar factors. As a consequence, the diagram transformations and notation for certain unitary operations involve "scalar gadgets" which denote contributions to a normalising factor. We present renormalised generators for the ZX calculus, which form a bialgebra precisely. As a result, no scalar gadgets are required to represent the most common unitary transformations, and the corresponding diagram transformations are generally simpler. We also present a similar renormalised version of the ZH calculus. We obtain these results by an analysis of conditions under which various "idealised" rewrites are sound, leveraging the existing presentations of the ZX and ZH calculi.