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We show how to find all $k$ marked elements in a list of size $N$ using theoptimal number $O(\sqrt{N k})$ of quantum queries and only a polylogarithmicoverhead in the gate complexity, in the setting where one has a small quantummemory. Previous algorithms either incurred a factor $k$ overhead in the gatecomplexity, 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 algorithmthat does so, with probability at least $1-\rho$, using $O(\sqrt{N \log(1/\rho)/ \delta})$ quantum queries (under mild assumptions on $\rho$). Thisquadratically improves the dependence on $1/\delta$ and $\log(1/\rho)$ comparedto a straightforward application of amplitude estimation. To obtain theimproved $\log(1/\rho)$ dependence we use the first result.

Basic quantum subroutines: finding multiple marked elements and summing numbers