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( Nk) 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 5-approximation of s = sigma Ni=1 vi where v = (vi) E [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 - p, using O(\iN log(1/p)/5) quantum queries (under mild assumptions on p). This quadratically improves the dependence on 1/5 and log(1/p) compared to a straightforward application of amplitude estimation. To obtain the improved log(1/p) dependence we use the first result.