Factoring 51 and 85 with 8 qubits

TL;DR

This paper presents simplified quantum circuits for factoring certain composite numbers, including 51 and 85, using only 8 qubits by exploiting their special algebraic properties, reducing quantum resource requirements.

Contribution

It introduces minimal quantum circuits for factoring composites with Fermat prime factors, significantly reducing the complexity of Shor's algorithm for these cases.

Findings

Factoring 51 and 85 requires only 8 qubits.

Modular exponentiation circuits use no more than four CNOT gates.

Simplified circuits leverage the property that the order modulo N is a power of 2.

Abstract

We construct simplified quantum circuits for Shor's order-finding algorithm for composites N given by products of the Fermat primes 3, 5, 17, 257, and 65537. Such composites, including the previously studied case of 15, as well as 51, 85, 771, 1285, 4369,... have the simplifying property that the order of a modulo N for every base a coprime to N is a power of 2, significantly reducing the usual phase estimation precision requirement. Prime factorization of 51 and 85 can be demonstrated with only 8 qubits and a modular exponentiation circuit consisting of no more than four CNOT gates.