Basic quantum subroutines: finding multiple marked elements and summing numbers

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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 d-approximation of s =ΣNi=1viwhere 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(√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.

Original languageEnglish
Article number1284
Number of pages29
Publication statusPublished - 2024

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