  This paper considers the conditional fault tolerance, $h$-super connectivity
$\kappa^{h}$ and $h$-super edge-connectivity $\lambda^{h}$ of the hierarchical
cubic network $HCN_n$, an attractive alternative network to the hypercube, and
shows $\kappa^h(HCN_n)=\lambda^h(HCN_n)=2^h(n+1-h)$ for any $h$ with $0\leq
h\leq n-1$. The results imply that at least $2^h(n+1-h)$ vertices or edges have
to be removed from $HCN_n$ to make it disconnected with no vertices of degree
less than $h$, and generalize some known results.
