  In this paper, we study the persistence approximation property for
quantitative $K$-theory of filtered $L^p$ operator algebras. Moreover, we
define quantitative assembly maps for $L^p$ operator algebras when $p\in
[1,\infty)$. Finally, in the case of $L^{p}$ crossed products and $L^{p}$ Roe
algebras, we find sufficient conditions for the persistence approximation
property. This allows us to give some applications involving the $L^{p}$
(coarse) Baum-Connes conjecture.
