let oracle $U_\psi$ prepare a state $\psi$ with success probability $p$. For simplicity assume that $U$ requires a single ancillary qubit and $\psi$ is itself a single-qubit state: $$ U_\psi |0\rangle|0\rangle = \sqrt{p}|0\rangle|\psi\rangle + \sqrt{1-p}|\bot\rangle \ . $$
Now I want to prepare a product state $|\psi\rangle|\psi\rangle|\psi\rangle\ldots$ ($m$ times). The most naive implementation requires $m$ ancillas, each of which has to be measured in the $|0\rangle$ state, which results in total success probability $p^m$ (exponentially small in $m$!). I could also reuse the same ancilla for all the oracles but I'm not sure if this changes anything.
This estimate seems very strange to me — I would naturally expect that there should exist an efficient way to prepare $|\psi\rangle|\psi\rangle|\psi\rangle\ldots$ Am I missing something obvious?
UPDATE
Note that applying amplitude amplification to each register does not resolve the issue since it can only increase the success probability polynomially. Amplitude amplification would lead to success probability exponentially close to 1 in the number of qubits per each $|\psi\rangle$ register, but I want to keep this number fixed and potentially small. I asked a related question here.
