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Register a new userNew factorization algorithm based on a continuous representation of truncated Gauss sums
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In this paper, we will describe a new factorization algorithm based on the continuous representation of Gauss sums, generalizable to orders j>2. Such an algorithm allows one, for the first time, to find all the factors of a number N in a single run without precalculating the ratio N/l, where l are all the possible trial factors. Continuous truncated exponential sums turn out to be a powerful tool for distinguishing factors from non-factors (we also suggest, with regard to this topic, to read an interesting paper by S. Woelk et al. also published in this issue [Woelk, Feiler, Schleich, J. Mod. Opt. in press]) and factorizing different numbers at the same time. We will also describe two possible M-path optical interferometers, which can be used to experimentally realize this algorithm: a liquid crystal grating and a generalized symmetric Michelson interferometer.
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We report the first implementation of a Gauss sum factorization algorithm by an internal state Ramsey interferometer using cold atoms. A sequence of appropriately designed light pulses interacts with an ensemble of cold rubidium atoms. The final population in the involved atomic levels determines a Gauss sum. With this technique we factor the number N=263193.
We study the theory of linearly chirped biphoton wave-packets produced in two basic quasi-phase-matching configurations: chirped photonic-like crystals and aperiodically poled crystals. The novelty is that these structures are considered as definite assembles of nonlinear layers that leads to detailed description of spontaneous parametric down-conversion (SPDC) processes through the discrete Gauss sums. We demonstrate that biphoton spectra for chirped photonic crystals involving a small number of layers consist from definite well-resolved spectral lines. We also discuss the forming of broadband spectra of signal (idler) waves in SPDC for both configurations as number of layers increases as well as in dependence of chirping parameters .
This letter introduces a real valued summation known as Complex Conjugate Pair Sum (CCPS). The space spanned by CCPS and its one circular downshift is called {em Complex Conjugate Subspace (CCS)}. For a given positive integer $Ngeq3$, there exists $frac{varphi(N)}{2}$ CCPSs forming $frac{varphi(N)}{2}$ CCSs, where $varphi(N)$ is the Eulers totient function. We prove that these CCSs are mutually orthogonal and their direct sum form a $varphi(N)$ dimensional subspace $s_N$ of $mathbb{C}^N$. We propose that any signal of finite length $N$ is represented as a linear combination of elements from a special basis of $s_d$, for each divisor $d$ of $N$. This defines a new transform named as Complex Conjugate Periodic Transform (CCPT). Later, we compared CCPT with DFT (Discrete Fourier Transform) and RPT (Ramanujan Periodic Transform). It is shown that, using CCPT we can estimate the period, hidden periods and frequency information of a signal. Whereas, RPT does not provide the frequency information. For a complex valued input signal, CCPT offers computational benefit over DFT. A CCPT dictionary based method is proposed to extract non-divisor period information.
To investigate a system coupled to a harmonic oscillator bath, we propose a new approach based on a phonon number representation of the bath. Compared to the method of the hierarchical equations of motion, the new approach is computationally much less expensive in a sense that a reduced density matrix is obtained by calculating the time evolution of vectors, instead of matrices, which enables one to deal with large dimensional systems. As a benchmark test, we consider a quantum damped harmonic oscillator, and show that the exact results can be well reproduced. In addition to the reduced density matrix, our approach also provides a link to the total wave function by introducing new boson operators.
Finding the factors of an integer can be achieved by various experimental techniques, based on an algorithm developed by Schleich et al., which uses specific properties of Gauss{}sums. Experimental limitations usually require truncation of these series, but if the truncation parameter is too small, it is no longer possible to distinguish between factors and so-called ghost factors. Here, we discuss two techniques for distinguishing between true factors and ghost factors while keeping the number of terms in the sum constant or only slowly increasing. We experimentally test these modified algorithms in a nuclear spin system, using NMR.
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