Basic quantum subroutines: finding multiple marked elements and summing numbers

Joran van Apeldoorn1, Sander Gribling2, and Harold Nieuwboer3

1IViR and QuSoft, University of Amsterdam, The Netherlands
2Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands
3Korteweg–de Vries Institute for Mathematics and QuSoft, University of Amsterdam, The Netherlands and Faculty of Computer Science, Ruhr University Bochum, Germany and Department of Mathematical Sciences, University of Copenhagen, Denmark

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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 $\delta$-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 algorithm that does so, with probability at least $1-\rho$, using $O(\sqrt{N \log(1/\rho) / \delta})$ quantum queries (under mild assumptions on $\rho$). This quadratically improves the dependence on $1/\delta$ and $\log(1/\rho)$ compared to a straightforward application of amplitude estimation. To obtain the improved $\log(1/\rho)$ dependence we use the first result.

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