Factorization of numbers with physical systems

Abstract

The periodicity properties of Gauss sums allow us to factor integer numbers. We show that the excitation probability amplitudes of appropriate quantum systems interacting with specific laser fields are determined by Gauss sums. The resulting probabilities are experimentally accessible by measuring the fluorescence from this level. In particular, we discuss a two‐photon transition in a ladder system driven by a chirped laser pulse. In addition, we consider two realizations of laser driven one‐photon transitions. For each quantum system we demonstrate the power of this factorization scheme using numerical examples.

Principles of quantum computing

“…Here we discuss a different algorithm based on Gauss sums [17]. An NMR implementation with proton spins (spin 1/2) demonstrates this algorithm by applying a series of phase gates in terms of phase shifted π-pulses [14].…”

Factorization in a single run with an optical interferometer

“…For j = 2, the truncated exponential sum reduces to a truncated Gauss sum. 6 If is a factor of N , all the terms interfere constructively and the modulo squared of the truncated exponential sum assumes its maximum value, i.e. 1.…”

Factorization of Numbers with Physical Systems

“…In the present section we summarize the key steps of Shor's algorithm 4 . The central idea bases on the connection of the factorization problem to the problem of determining the order r in modular exponentiation n r mod N = 1,…”