Wave Packets Can Factorize Numbers

Abstract

We draw attention to various aspects of number theory emerging in the time evolution of elementary quantum systems with quadratic phases. Such model systems can be realized in actual experiments. Our analysis paves the way to a new, promising and effective method to factorize numbers. IntroductionInterference of waves instead of Newton's light corpuscles --what a revolutionary idea of Thomas Young pronounced in his famous talk at the Royal Society in 1801 [1]. The opposite idea --particles as waves --proposed by Louis de Broglie and cast into the appropriate mathematical form by Erwin Schrö dinger more than 100 years later opened a new realm in atomic physics with more applications today than ever. ''Can a quantum-mechanical description of physical reality be considered complete?" This famous question raised by Albert Einstein, Boris Podolsky and Nathan Rosen in their seminal paper [2] brought out most clearly the importance of entanglement [3] as the basic ingredient of quantum theory. Interference, wave mechanics and entanglement are the three ideas on which Peter Shor could build his factorization algorithm [4] with a polynomial rather than exponential speed. This discovery gave birth to the new field of quantum information [5]. Guided by the same three central ideas we show in the present paper that the time evolution of elementary quantum systems carries intrinsically the tools to factorize numbers [6].Stationary states have played an important role in the development of quantum mechanics. For this reason the time evolution of wave packets has for many years taken a back seat even in textbooks on quantum theory. In this context it is interesting to recall that already in 1927 Kennard [7] studied the time evolution of wave packets in simple potentials [8]. Recently, sophisticated techniques to create short laser pulses have opened a new era for a controlled creation and observation of wave packets in atoms [9], molecules [10] and cavities [11]. Even wave packet dynamics of cold atoms [12] and Bose-Einstein condensates [13] has been observed. These examples illustrate convincingly that today's technology allows to promote this gedankenexperiment of factorization to a real implementation in a quantum system.The use of wave packet dynamics to factorize numbers has been recognized for special examples. Clauser and Dowling [14] uses a N-slit Young interference setup. Hence, the number to be factorized is encoded in the number of slits. The indicator of a factor is the interference pattern on a screen as a function of the wavelength of the light or matter wave. Whenever the intensity across the screen is constant, the appropriately scaled wavelength provides a factor. Similarly, Harter [15] considers the time evolution of a particle in a box. In contrast to these contributions we provide a general framework for factorization and moreover, exhaust the potential of interference by capitalizing on properties of quadratic phase factors.

Fractal noise in quantum ballistic and diffusive lattice systems

“…In fact, P 0 (t) can reveal the prime factors of an encoded number. 12 On the other hand, fractals known from nonlinear physics also appear in quantum physics, despite the fact that quantum mechanics is linear. The space fractal dimension D x ϭ1.5 obtained for the spatial distribution corresponds to Brownian 1/f 2 noise familiar from random walks.…”

Quantum wave packet revivals

“…Averbukh and Perelman (34) were the first to analyze in detail the mathematical structure of the additional phase factors arising from the exp(−2πik 2 t/T rev ) terms at such times to discuss the "Universality in the long-term evolution of quantum wave packets beyond the correspondence principle limit" and we reproduce here, in part, their elegant arguments, for completeness. (We note that it has been pointed out (63), (64) that this problem, especially the calculation of the autocorrelation function at fractional revival times, is similar to that of the evaluation of Gauss sums (65) which has a long history in the mathematical literature (66).) The case of odd q is most straightforward and we consider the more general arguments for even q in Appendix C.…”

Quantum phase measurement and Gauss sum factorization of large integers in a superconducting circuit

“…The phase factor cos(ωτ k ) can be determined from the probability of the excited state of qubit 1 in Eq. (29). By repeating the same procedure K + 1 times and setting the time duration τ k for k = 0, .…”