We say a graph is (Q
n
,Q
m
)-saturated if it is a maximal Q
m
-free subgraph of the n-dimensional hypercube Q
n
. A graph is said to be (Q
n
,Q
m
)-semi-saturated if it is a subgraph of Q
n
and adding any edge forms a new copy of Q
m
. The minimum number of edges a (Q
n
,Q
m
)-saturated graph (respectively (Q
n
,Q
m
)-semi-saturated graph) can have is denoted by sat(Q
n
,Q
m
) (respectively s-sat(Q
n
,Q
m
)). We prove that
$$
\begin{linenomath}
\lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0,
\end{linenomath}$$
for fixed
m, disproving a conjecture of Santolupo that, when
m=2, this limit is 1/4. Further, we show by a different method that sat(
Q
n
,
Q
2)=
O(2
n
), and that
s-sat(
Q
n
,
Q
m
)=
O(2
n
), for fixed
m. We also prove the lower bound
$$
\begin{linenomath}
\ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n,
\end{linenomath}$$
thus determining sat(
Q
n
,
Q
2) to within a constant factor, and discuss some further questions.