  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.
