title: Quantum Simulation of the Factorization Problem.
abstract: Feynman's prescription for a quantum simulator was to find a Hamitonian for a system that could serve as a computer. The Pólya-Hilbert conjecture proposed the demonstration of Riemann's hypothesis through the spectral decomposition of Hermitian operators. Here we study the problem of decomposing a number into its prime factors, N=xy, using such a simulator. First, we derive the Hamiltonian of the physical system that simulates a new arithmetic function formulated for the factorization problem that represents the energy of the computer. This function rests alone on the primes below sqrt[N]. We exactly solve the spectrum of the quantum system without resorting to any external ad hoc conditions, also showing that it obtains, for x≪sqrt[N], a prediction of the prime counting function that is almost identical to Riemann's R(x) function. It has no counterpart in analytic number theory, and its derivation is a consequence of the quantum theory of the simulator alone.
authors.affiliation: Center for Computational Simulation, ETS Ingenieros Informáticos, Universidad Politécnica de Madrid, Campus Montegancedo, E28660 Madrid, Spain.
authors.forename: Jose Luis
authors.initials: JL
authors.lastname: Rosales
authors.affiliation: Center for Computational Simulation, ETS Ingenieros Informáticos, Universidad Politécnica de Madrid, Campus Montegancedo, E28660 Madrid, Spain.
authors.forename: Vicente
authors.initials: V
authors.lastname: Martin