Quantum amplitude estimation

Amplitude estimation extracts the amplitude $a$ hidden in a state $\mathcal A|0\rangle=\sqrt{1-a}\,|0\rangle|\phi_0\rangle+\sqrt a\,|1\rangle|\phi_1\rangle$ prepared by a state-preparation operator $\mathcal A$, where $a$ is typically a success probability or expectation of interest. The original construction runs phase estimation on the Grover-style operator $Q=-\mathcal A S_0 \mathcal A^\dagger S_\chi$ built from $\mathcal A$ and two reflections; $Q$'s eigenphases encode $a=\sin^2\theta$, so measuring $m$ phase-register qubits yields an estimate $\hat a$ with error scaling as $O(2^{-m})=O(1/\varepsilon)$ queries to $\mathcal A$, compared with $O(1/\varepsilon^2)$ independent classical samples needed for the same error via a Chernoff/Hoeffding bound — a quadratic query reduction assuming coherent access to $\mathcal A$. This record checks a small Bernoulli-amplitude toy circuit against repeated classical sampling; the query-count advantage, as the entry's caveat states, depends entirely on state-preparation and oracle-access costs a toy circuit cannot certify.