Modified Iterative Quantum Amplitude Estimation is Asymptotically Optimal

Abstract: In this work, we provide the first QFT-free algorithm for Quantum Amplitude Estimation (QAE) that is asymptotically optimal while maintaining the leading numerical performance. QAE algorithms appear as a subroutine in many applications for quantum computers. The optimal query complexity achievable by a quantum algorithm for QAE is $O\left(\frac{1}{\epsilon}\log \frac{1}{\alpha}\right)$ queries, providing a speedup of a factor of $1/\epsilon$ over any other classical algorithm for the same problem. The original algorithm for QAE utilizes the quantum Fourier transform (QFT) which is expected to be a challenge for near-term quantum hardware. To solve this problem, there has been interest in designing a QAE algorithm that avoids using QFT. Recently, the iterative QAE algorithm (IQAE) was introduced by Grinko et al. with a near-optimal $O\left(\frac{1}{\epsilon}\log \left(\frac{1}{\alpha} \log \frac{1}{\epsilon}\right)\right)$ query complexity and small constant factors. In this work, we combine ideas from the preceding line of work to introduce a QFT-free QAE algorithm that achieves a query complexity upper bound of $\frac{62}{\epsilon}\ln\frac{6}{\alpha}$ which matches the optimal $O\left(\frac{1}{\epsilon}\log \frac{1}{\alpha}\right)$ query complexity. We supplement our analysis with numerical experiments comparing our performance with IQAE where we find that our modifications retain the high performance, and in some cases even improve the numerical results.