  The restricted $h$-connectivity of a graph $G$, denoted by $\kappa^h(G)$, is
defined as the minimum cardinality of a set of vertices $F$ in $G$, if exists,
whose removal disconnects $G$ and the minimum degree of each component of $G-F$
is at least $h$. In this paper, we study the restricted $h$-connectivity of the
balanced hypercube $BH_n$ and determine that
$\kappa^1(BH_n)=\kappa^2(BH_n)=4n-4$ for $n\geq2$. We also obtain a sharp upper
bound of $\kappa^3(BH_n)$ and $\kappa^4(BH_n)$ of $n$-dimension balanced
hypercube for $n\geq3$ ($n\neq4$). In particular, we show that
$\kappa^3(BH_3)=\kappa^4(BH_3)=12$.
