  The main purpose of this paper is to show that we can exploit the difference
($l_1$-norm and $l_2$-norm) in the probability calculation between quantum and
probabilistic computations to claim the difference in their space efficiencies.
It is shown that there is a finite language $L$ which contains sentences of
length up to $O(n^{c+1})$ such that: ($i$) There is a one-way quantum finite
automaton (qfa) of $O(n^{c+4})$ states which recognizes $L$. ($ii$) However, if
we try to simulate this qfa by a probabilistic finite automaton (pfa)
\textit{using the same algorithm}, then it needs $\Omega(n^{2c+4})$ states. It
should be noted that we do not prove real lower bounds for pfa's but show that
if pfa's and qfa's use exactly the same algorithm, then qfa's need much less
states.
