Prime factorization of arbitrary integers with a logarithmic energy spectrum

Abstract

We propose an iterative scheme to factor numbers based on the quantum dynamics of an ensemble of interacting bosonic atoms stored in a trap where the single-particle energy spectrum depends logarithmically on the quantum number. When excited by a time-dependent interaction these atoms perform Rabi oscillations between the ground state and an energy state characteristic of the factors. The number to be factored is encoded into the frequency of the sinusoidally modulated interaction. We show that a measurement of the energy of the atoms at a time chosen at random yields the factors with probability one half. We conclude by discussing a protocol to obtain the desired prime factors employing a logarithmic energy spectrum which consists of prime numbers only.

Holographic realization of the prime number quantum potential

“…the potential that has the first N lucky numbers as eigenvalues, and we experimentally realized the case N = 10. Work is in progress to realize quantum potentials which have as energy spectra sequences such as the Fibonacci numbers ( 53 ), or the real sequence of the logarithms of the integers, or of the logarithms of the primes, as considered in ( 15 , 16 , 54 ). Indeed, as we have already emphasized, any finite sequence of integer or real numbers can be obtained.…”

“…Given the role played by prime numbers in many problems, from factorization of integers to celebrated conjectures of mathematics such as the Riemann hypothesis ( 10–12 ) or the Goldbach conjecture ( 13 ), it is desirable to find new ways in which prime numbers emerge from the experimental control of a quantum system. This is particularly important in view of several theoretical proposals to tackle problems from number theory using quantum mechanics ideas: prominent examples include Shor’s algorithm ( 14 ), the factorization of large integers ( 15 , 16 ), the computation of prime number functions by employing the so-called quantum prime state ( 17 , 18 ), primality tests ( 19 , 20 ), and attempts to establish the validity of the Riemann hypothesis [see refs. ( 21 , 22 ) for reviews].…”

Factorization with a Logarithmic Energy Spectrum of a Central Potential

“…As an alternative method, we have studied [4][5][6] the factorization of integers using bosonic atoms in one-and two-dimensional potentials, both with a logarithmic energy spectrum. Our present theoretical study represents an extension of these thoughts and is motivated by two features: (i) it is possible [7] to create and control almost any kind of potential for the center-of-mass motion of the atom using adiabatic potentials, and (ii) bosons in a spherically symmetric parabolic potential as well as in a spherical box provide textbook examples for the thermodynamics of the Bose-Einstein condensation [8,9].…”

Factorization with a logarithmic energy spectrum of a central potential

“…As an alternative method we have studied the factorization of integers using bosonic atoms in one-and two-dimensional potentials both with a logarithmic energy spectrum. [4][5][6] Bosons in a spherically symmetric harmonic potential as well as in a spherical box provide textbook examples for the study of thermodynamics of the Bose-Einstein condensation. [7,8] Our present theoretical study is motivated by the possibility to create and control nearly any kind of traps using adiabatic potentials as was stated in Ref.…”