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 δ-approximation of s=∑i=1Nvi where v=(vi)∈[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(Nlog⁡(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.