An efficient quantum algorithm for the Moebius function

TL;DR

This paper presents a quantum algorithm that efficiently computes the Moebius function for natural numbers without needing prime factorization, with complexity quadratic in the logarithm of the input size.

Contribution

It introduces a novel quantum algorithm for the Moebius function that is more efficient than classical methods and avoids prime factorization.

Findings

Algorithm runs in asymptotically quadratic time in log n

Does not require prime factorization of n

Provides a faster quantum approach for Moebius function computation

Abstract

We give an efficient quantum algorithm for the Moebius function $\mu(n)$ from the natural numbers to $\{-1,0,1\}$. The cost of the algorithm is asymptotically quadratic in $\log n$ and does not require the computation of the prime factorization of $n$ as an intermediate step.