In 1998, Brassard, Hoyer, Mosca, and Tapp (BHMT) gave a quantum algorithm for approximate counting. Given a list of $N$ items, $K$ of them marked, their algorithm estimates $K$ to within relative error $\varepsilon$ by making only $O\left( \frac{1}{\varepsilon}\sqrt{\frac{N}{K}}\right) $ queries. Although this speedup is of "Grover" type, the BHMT algorithm has the curious feature of relying on the Quantum Fourier Transform (QFT), more commonly associated with Shor's algorithm. Is this necessary? This paper presents a simplified algorithm, which we prove achieves the same query complexity using Grover iterations only. We also generalize this to a QFT-free algorithm for amplitude estimation. Related approaches to approximate counting were sketched previously by Grover, Abrams and Williams, Suzuki et al., and Wie (the latter two as we were writing this paper), but in all cases without rigorous analysis.

翻译：1998年,Brassard、Hoyer、Mosca和Tapp(BHMT)给出了一种量子算法。根据一份以美元计价的项目清单,其中标注了美元,他们的算法估计美元在相对差错范围内为美元 $varepsilon$, 仅用折叠式转折法将美元估算为美元。虽然这种加速是“Grover”类型,但BHMT算法具有依赖量子Fourier变换法(QFT)这一奇特特征,后者通常与Shor的算法有关。 是否有必要? 本文提出了一个简化的算法, 我们证明它只使用折叠式转式转折法就能达到相同的查询复杂程度。 我们还将其概括为“ QFT” 缩略式算法。 与估算相近似的算法此前由 Grover、 Abrams 和 Williams、 Suzuki et al. 和 Wie (我们正在撰写的后两部), 但在所有情况下都没有经过严格的分析。