Quantum Approximate Counting, Simplified

TL;DR

This paper introduces a simplified quantum approximate counting algorithm that matches the original's query complexity without using the Quantum Fourier Transform, relying solely on Grover iterations, and extends it to a QFT-free amplitude estimation method.

Contribution

It provides a rigorously analyzed, QFT-free quantum approximate counting algorithm that simplifies previous methods and extends to amplitude estimation.

Findings

Achieves the same query complexity as the original BHMT algorithm

Eliminates the need for Quantum Fourier Transform

Provides a rigorous analysis of the simplified algorithm

Abstract

In 1998, Brassard, Hoyer, Mosca, and Tapp (BHMT) gave a quantum algorithm for approximate counting. Given a list of $N$ items, $K$ of them marked, their algorithm estimates $K$ to within relative error $\varepsilon$ by making only $O\left( \frac{1}{\varepsilon}\sqrt{\frac{N}{K}}\right) $ queries. Although this speedup is of "Grover" type, the BHMT algorithm has the curious feature of relying on the Quantum Fourier Transform (QFT), more commonly associated with Shor's algorithm. Is this necessary? This paper presents a simplified algorithm, which we prove achieves the same query complexity using Grover iterations only. We also generalize this to a QFT-free algorithm for amplitude estimation. Related approaches to approximate counting were sketched previously by Grover, Abrams and Williams, Suzuki et al., and Wie (the latter two as we were writing this paper), but in all cases without rigorous analysis.