Basic quantum subroutines: finding multiple marked elements and summing numbers

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 $δ$-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 $1-ρ$, using $O(\sqrt{N \log(1/ρ) / δ})$ quantum queries (under mild assumptions on $ρ$). This quadratically improves the dependence on $1/δ$ and $\log(1/ρ)$ compared to a straightforward application of amplitude estimation. To obtain the improved $\log(1/ρ)$ dependence we use the first result.