In 2009 Galbraith and Lin wrote the article "Computing Pairings Using x-Coordinates Only" https://link.springer.com/article/10.1007/s10623-008-9233-3, where they proposed to compute pairings on elliptic curves in the compressed form, i.e., to use only $x$-coordinates of points. This is similar to the well known efficient Montgomery exponentiation. However, in the end of the article the authors remarked that their approach is not efficient for large embedding degrees such as $k=12$ for popular Barreto-Naehrig curves.
What is the state of the art today? The ideas of Galbraith and Lin have been improved and implemented in practice? Or pairings on Barreto-Naehrig curves are computed immediately on elliptic curves, i.e., using equally $x$ and $y$ coordinates as in the classical work of Miller?