  Let ${\mathscr{C}}$ be an $n$-cluster tilting subcategory of an exact
category $({\mathscr{A}}, {\mathscr{E}})$, where $n \geq 1$ is an integer. It
is proved by Jasso that if $n> 1$, then ${\mathscr{C}}$ although is no longer
exact, but has a nice structure known as $n$-exact structure. In this new
structure conflations are called admissible $n$-exact sequences and are
${\mathscr{E}}$-acyclic complexes with $n+2$ terms in ${\mathscr{C}}$. Since
their introduction by Iyama, cluster tilting subcategories has gained a lot of
traction, due largely to their links and applications to many research areas,
many of them unexpected. On the other hand, ideal approximation theory, that is
a gentle generalization of the classical approximation theory and deals with
morphisms and ideals instead of objects and subcategories, is an active area
that has been the subject of several researches. Our aim in this paper is to
introduce the so-called `ideal approximation theory' into `higher homological
algebra'. To this end, we introduce some important notions in approximation
theory into the theory of $n$-exact categories and prove some results. In
particular, the higher version of the notions such as ideal cotorsion pairs,
phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals will be
introduced and studied. Our results motivate the definitions and show that
$n$-exact categories are the appropriate context for the study of `higher ideal
approximation theory'.
