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内存的设置下，利用最优的量子查询次数 $O(\sqrt{N k})$ 找到大小为 $N$ 的列表中的所有 $k$ 个标记元素，且门复杂度仅需多对数量级开销。此前算法要么在门复杂度上引入 $k$ 倍的额外开销，要么在查询复杂度上增加额外的 $\log(k)$ 因子。接着，我们考虑了对 $s = \sum_{i=1}^N v_i$（其中 $v=(v_i) \in [0,1]^N$）进行乘法 $\delta$ 近似的问题，给定对 $v$ 的二进制描述的量子查询访问。我们提出了一种算法，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）。与直接应用振幅估计相比，这将 $1/\delta$ 和 $\log(1/\rho)$ 的依赖关系进行了二次改进。为获得改进的 $\log(1/\rho)$ 依赖关系，我们使用了第一个结果。