Integer Factorization by Quantum Measurements

TL;DR

This paper introduces a novel quantum measurement-based algorithm for integer factorization that efficiently factors numbers in a number of steps equal to their prime factors, potentially reaching fundamental lower bounds.

Contribution

It presents a new quantum algorithm leveraging quantum measurement, differing from Shor's algorithm, and achieves factorization in minimal steps related to the number of prime factors.

Findings

Factorization in steps equal to the number of prime factors

Quantum measurement saturates the lower bound of operations

Efficient for numbers with few prime factors

Abstract

Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as entanglement and superposition of states. Among the known quantum algorithms, a special role is played by the Shor algorithm, i.e. a polynomial-time quantum algorithm for integer factorization, with far reaching potential applications in several fields, such as cryptography. Here we present a different algorithm for integer factorization based on another genuine quantum property: quantum measurement. In this new scheme, the factorization of the integer $N$ is achieved in a number of steps equal to the number $k$ of its prime factors, -- e.g., if $N$ is the product of two primes, two quantum measurements are enough, regardless of the number of digits $n$ of the number $N$. Since $k$ is the lower bound to the number of operations one can do to factorize a general integer, one sees that a quantum mechanical setup can saturate such a bound.