# Simpler Quantum Counting

## TL;DR

This paper introduces a simplified quantum counting algorithm that uses amplitude amplification, requiring fewer quantum resources and providing faster, more accurate estimates of the number of marked states compared to previous methods, especially when the ratio M/N is small.

## Contribution

The paper presents a new quantum counting algorithm based on amplitude amplification that is simpler, more resource-efficient, and faster than phase estimation-based algorithms for small M/N ratios.

## Key findings

- Requires fewer quantum resources in circuit width and depth.
- Produces more accurate estimates of M.
- Runs significantly faster when M/N is small.

## Abstract

A simpler quantum counting algorithm based on amplitude amplification is presented. This algorithm is bounded by O(sqrt(N/M)) calls to the controlled-Grover operator where M is the number of marked states and N is the total number of states in the search space. This algorithm terminates within log(sqrt(N/M)) consecutive measurement steps when the probability p1 of measuring the state |1> is at least 0.5, and the result from the final step is used in estimating M by a classical post processing. The purpose of controlled-Grover iteration is to increase the probability p1. This algorithm requires less quantum resources in terms of the width and depth of the quantum circuit, produces a more accurate estimate of M, and runs significantly faster than the phase estimation-based quantum counting algorithm when the ratio M/N is small. We compare the two quantum counting algorithms by simulating various cases with a different M/N ratio, such as M/N > 0.125 or M/N < 0.001.

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Source: https://tomesphere.com/paper/1907.08119