Adaptive Algorithm for Quantum Amplitude Estimation

Abstract

Quantum amplitude estimation is a key sub-routine of a number of quantum algorithms with various applications. We propose an adaptive algorithm for interval estimation of amplitudes. The quantum part of the algorithm is based only on Grover's algorithm. The key ingredient is the introduction of an adjustment factor, which adjusts the amplitude of good states such that the amplitude after the adjustment, and the original amplitude, can be estimated without ambiguity in the subsequent step. We show with numerical studies that the proposed algorithm uses a similar number of quantum queries to achieve the same level of precision ǫ compared to state-of-the-art algorithms, but the classical part, i.e., the non-quantum part, has substantially lower computational complexity. We rigorously prove that the number of oracle queries achieves O(1/ǫ), i.e., a quadratic speedup over classical Monte Carlo sampling, and the computational complexity of the classical part achieves O(log(1/ǫ)), both up to a double-logarithmic factor.

Quantum Monte Carlo algorithm for solving Black-Scholes PDEs for high-dimensional option pricing in finance and its proof of overcoming the curse of dimensionality

“…It is noted that Brassard et al's algorithm enables a quadratic speed-up for many approximation problems which are solved classically by Monte Carlo simulations under the assumption that the corresponding distribution can be uploaded in rotated form (41). However, due to difficulties in implementing large number of controlled unitary operators as well as the quantum Fourier transform (QFT) operator on quantum computers, several variants of the QAE algorithm without using QFT have been proposed recently; see [1,29,48,49,58,61,63]. In this paper, we use the modified iterative quantum amplitude estimation algorithm (Modified IQAE) [27, Algorithm 1] introduced recently by Fukuzawa et al [27], which is a modification of the IQAE algorithm presented by Suzuki et al [58].…”

“…Several variations of the Quantum Amplitude Estimation algorithm have been proposed recently, see e.g. [1,27,29,48,49,58,61,63]. Quantum Monte Carlo methods can ideally achieve a quadratic speed-up [34], [42] compared to classical (i.e.…”

Amplitude Estimation from Quantum Signal Processing

“…[Nakaji20] claims to present a modified version of IQAE with better query complexity, but the experiments in the manuscript do not support this claim 5 . Further, [Zhao&22] gave an algorithm where the classical processing is significantly faster then IQAE, but the query complexity is about the same.…”

Amplitude Estimation from Quantum Signal Processing

“…[Nakaji20] claims to present a modified version of IQAE with better query complexity, but the experiments in the manuscript do not support this claim 2 . Further, [Zhao&22] gave an algorithm where the classical processing is significantly faster then IQAE, but the query complexity is about the same.…”