Asymptotic Quantum Search and a Quantum Algorithm for Calculation of a Lower Bound of the Probability of Finding a Diophantine Equation That Accepts Integer Solutions

TL;DR

This paper introduces asymptotic quantum search, a novel quantum algorithm that finds function minima with a single measurement, and applies it to estimate a lower bound on the probability of solving Diophantine equations with quantum counting.

Contribution

It presents a new quantum algorithm that avoids intermediate measurements and subroutines, improving efficiency for certain search problems.

Findings

The algorithm requires only one measurement, unlike previous methods.

It can estimate a lower bound on the probability of finding solutions to Diophantine equations.

Demonstrates potential for more efficient quantum search techniques.

Abstract

Several mathematical problems can be modeled as a search in a database. An example is the problem of finding the minimum of a function. Quantum algorithms for solving this problem have been proposed and all of them use the quantum search algorithm as a subroutine and several intermediate measurements are realized. In this work, it is proposed a new quantum algorithm for finding the minimum of a function in which quantum search is not used as a subroutine and only one measurement is needed. This is also named asymptotic quantum search. As an example, we propose a quantum algorithm based on asymptotic quantum search and quantum counting able to calculate a lower bound of the probability of finding a Diophantine equation with integer solution.