Nearly Optimal Amplitude Estimation at any Depth
We develop a class of amplitude estimation algorithms with tunable circuit depth $M$ and circuit repetitions $N$, requiring neither ancilla qubits nor controlled Grover operations. For additive error $\epsilon$ in the Grover angle, they attain the nearly optimal query-depth tradeoff $M^2N\in\widetilde{O}(\epsilon^{-2})$ uniformly over $\lambda\in[0,\pi/2]$, spanning the full range from classical sampling at $M=1$ to the Heisenberg limit at $M=\Theta(\epsilon^{-1})$. While prior depth-tunable work establishes comparable angle-accuracy guarantees only away from the boundaries $\lambda\to0$ or $\pi/2$ or at discrete depth tradeoffs, our guarantee extends to both boundaries, so the quantum speedup persists there rather than degrading to classical sampling. Numerical experiments confirm the predicted uniform angle accuracy and show low overhead in practice, making them strong candidates for practical amplitude estimation in the early fault-tolerant regime, with applications such as overlap certification, trial-state verification, and Monte Carlo methods.