Decompose stochastic matrix in product of two stochastic matrices

Question

There exist stochastic matrices $Q$ such that there is no stochastic matrix $P$ such that $P^2=Q$.

I am interested in the following problem:

For a given stochastic matrix $Q$, find stochastic matrices $P_1$ and $P_2$ that minimize $\|P_ 1-P_2\|$ such that $P_1 P_2 = Q$.

Does anybody know of references or has ideas on how to tackle this?