Can we use quantum machines to reduce space complexity of deterministic turing machines?

Question

Can we convert every algorithm in $\text{P}$ (polynomial time complexity for deterministic machines) into a quantum algorithm with polynomial time and $O(\log n)$ quantum bit?

Answer 1

It seems this problem is open.

Watrous [J. Comp. Sys. Sci. 59, (pp. 281-326), 1999] proved that any space $s$ bounded quantum Turing Machine (for space constructible $s(n)>\Omega(\log n)$) can be simulated by deterministic Turing machine with $O(s^2)$ space. With the assumption $\mathsf{P \neq SC}$ (where $\mathsf{SC \subseteq P}$ is defined as the class of problems solvable by a DTM simultaneously in polynomial time and poly-logarithmic space), quantum machines will not reduce space complexity exponentially.

N.B. We don't know whether $\mathsf{P=SC}$ or not, though it is considered unlikely that they would be equal.