We show how to find all $k$ marked elements in a list of size $N$ using the optimal number $O(\sqrt{N k})$ of quantum queries and only a polylogarithmic overhead in the gate complexity, in the setting where one has a small quantum memory. Previous algorithms either incurred a factor $k$ overhead in the gate complexity, or had an extra factor $\log(k)$ in the query complexity. We then consider the problem of finding a multiplicative $\delta$-approximation of $s = \sum_{i=1}^N v_i$ where $v=(v_i) \in [0,1]^N$, given quantum query access to a binary description of $v$. We give an algorithm that does so, with probability at least $1-\rho$, using $O(\sqrt{N \log(1/\rho) / \delta})$ quantum queries (under mild assumptions on $\rho$). This quadratically improves the dependence on $1/\delta$ and $\log(1/\rho)$ compared to a straightforward application of amplitude estimation. To obtain the improved $\log(1/\rho)$ dependence we use the first result.

翻译：我们展示了如何在小量子存储器设置下，使用最优量子查询次数$O(\sqrt{N k})$且门复杂度仅需多对数开销，在大小为$N$的列表中找出所有$k$个标记元素。此前算法要么在门复杂度上带来因子$k$的额外开销，要么在查询复杂度上增加因子$\log(k)$。随后我们考虑对$s = \sum_{i=1}^N v_i$（其中$v=(v_i) \in [0,1]^N$）寻找乘法$\delta$-近似值的问题，假设可通过量子查询访问$v$的二进制描述。我们提出一种算法，在概率至少为$1-\rho$的情况下，使用$O(\sqrt{N \log(1/\rho) / \delta})$次量子查询（在$\rho$的温和假设下）。相比于直接应用振幅估计，该结果在$1/\delta$和$\log(1/\rho)$的依赖关系上实现了二次改进。为获得改进的$\log(1/\rho)$依赖关系，我们使用了第一个结果。