Basic quantum subroutines: finding multiple marked elements and summing numbers

TL;DR

This paper presents optimal quantum algorithms for finding multiple marked elements with minimal query complexity and for approximating sums of bounded values, improving efficiency over previous methods.

Contribution

It introduces quantum algorithms that achieve optimal query complexity for finding multiple marked elements and for sum approximation, with reduced overhead and improved probabilistic dependence.

Findings

Optimal quantum query complexity for finding all k marked elements: O(√N k).

Quantum sum approximation algorithm with improved dependence on δ and ρ.

Polylogarithmic overhead in gate complexity for marked elements problem.

Abstract

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.