Bell states in a Penning trap as the quantum simulator of the integer factorization problem

The recently introduced equivalent formulation of the integer factorization problem for N = xy, obtaining a function of the primes E(x) within the factorization ensemble, is reviewed. Here we demonstrate that this formulation can be readily translated to the physics of a quantum device in which the quantities Ek are the eigenvalues of a bounded Hamiltonian. The spectrum is solved for xk = o( √ N) leading to a super efficient probabilistic quantum factoring algorithm which only requires o((log √ N)) measurements. The state of the quantum simulator can be identified as that of a two electrons P wave in a Penning Trap. We consider the possibility to build the simulator experimentally in order to obtain, from the measured magnetron trap frequencies, the probabilistic quantum sieve for the possible factors of N . This approach is suited for large N and x = o( √ N).