Accelerated Quantum Amplitude Estimation without QFT

We put forward a Quantum Amplitude Estimation algorithm delivering superior performance (lower quantum computational complexity and faster classical computation parts) compared to the approaches available to-date. The algorithm does not relay on the Quantum Fourier Transform and its quantum computational complexity is of order $O(\frac{1}{\varepsilon})$ in terms of the target accuracy $\varepsilon>0$. The $O(\frac{1}{\varepsilon})$ bound on quantum computational complexity is also superior compared to those in the earlier approaches due to smaller constants. Moreover, a much tighter bound is obtained by means of computer-assisted estimates for the expected value of quantum computational complexity. The correctness of the algorithm and the $O(\frac{1}{\varepsilon})$ bound on quantum computational complexity are supported by precise proofs.