 Solution

Apply Hadamard gates to $|0^m⟩$ to prepareReversibly compute the efficiently decidable predicate $[x<Q]$ into an ancillary qubit and measure it. The success probability is $Q/2^m>1/2$, and conditioned on success the first register is exactlyRestart after a failed measurement. After $L=\lceil {\log }_2(1/\delta )\rceil$ attempts, the probability that all attempts fail is below $\delta$, so the state is prepared with probability at least $1-\delta$ using a number of gates polynomial in $m$ and $\log (1/\delta )$.