title: Factoring 51 and 85 with 8 qubits.
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.
authors.affiliation: Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA.
authors.forename: Michael R
authors.initials: MR
authors.lastname: Geller
authors.forename: Zhongyuan
authors.initials: Z
authors.lastname: Zhou