HUBO & QUBO and Prime Factorization

This document details the methodology and steps taken to convert Higher Order Unconstrained Binary Optimization (HUBO) models into Quadratic Unconstrained Binary Optimization (QUBO) models. The focus is primarily on prime factorization problems; a critical and computationally intensive task relevant in various domains including cryptography, optimization, and number theory. The conversion from Higher-Order Binary Optimization (HUBO) to Quadratic Unconstrained Binary Optimization (QUBO) models is crucial for harnessing the capabilities of advanced computing methodologies, particularly quantum computing and DYNEX neuromorphic computing. Quantum computing offers potential exponential speedups for specific problems through its intrinsic parallelism capabilities. Conversely, DYNEX neuromorphic computing enhances efficiency and accelerates the resolution of intricate, pattern-oriented tasks by simulating memristors in GPUs, employing a highly decentralized approach, via Blockchain technology. This transformation enables the exploitation of these cutting-edge computing paradigms to address complex optimization challenges effectively. Through detailed explanations, mathematical formulations, and algorithmic strategies, this document aims to provide a comprehensive guide to understanding and implementing the conversion process from HUBO to QUBO. It underscores the importance of such transformations in making prime factorization computationally feasible on both existing classical computers and emerging computing technologies

A new method to determine X-ray luminosity functions of AGN and their evolution with redshift

Almost all massive galaxies today are understood to contain supermassive black holes (SMBH) at their centres. SMBHs grew by accreting material from their surroundings, emitting X-rays as they did so. X-ray luminosity functions (XLFs) of active galactic nuclei (AGN) have been extensively studied in order to understand the AGN population’s cosmological properties and evolution. We present a new fixed rest-frame method to achieve a more accurate study of the AGN XLF evolution over cosmic time. Normally, XLFs are constructed in a fixed observer-frame energy band, which can be problematic because it probes different rest-frame energies at different redshifts. In the new method, we construct XLFs in the fixed rest-frame band instead, by varying the observed energy band with redshift. We target a rest-frame 2–8 keV band using XMM-Newton and HEAO 1 X-ray data, with seven observer-frame energy bands that vary with redshift for 0 < z < 3. We produce the XLFs using two techniques; one to construct a binned XLF, and one using a maximum likelihood (ML) fit, which makes use of the full unbinned source sample. We find that our ML best-fitting pure luminosity evolution results for both methods are consistent with each other, suggesting that performing XLF evolution studies with the high-redshift data limited to high-luminosity AGN is not very sensitive to the choice of fixed observer-frame or rest-frame energy band, which is consistent with our expectation that high-luminosity AGN typically show little ABSORPTION. We have demonstrated the viability of the new method in measuring the XLF evolution

A New Method for TSVD Regularization Truncated Parameter Selection

The truncated singular value decomposition (TSVD) regularization applied in ill-posed problem is studied. Through mathematical analysis, a new method for truncated parameter selection which is applied in TSVD regularization is proposed. In the new method, all the local optimal truncated parameters are selected first by taking into account the interval estimation of the observation noises; then the optimal truncated parameter is selected from the local optimal ones. While comparing the new method with the traditional generalized cross-validation (GCV) and L curve methods, a random ill-posed matrices simulation approach is developed in order to make the comparison as statistically meaningful as possible. Simulation experiments have shown that the solutions applied with the new method have the smallest mean square errors, and the computational cost of the new algorithm is the least

Full Counting Statistics of Interacting Electrons

In order to fully characterize the noise associated with electron transport, with its severe consequences for solid-state quantum information systems, the theory of full counting statistics has been developed. It accounts for correlation effects associated with the statistics and effects of entanglement, but it remains a non-trivial task to account for interaction effects. In this article we present two examples: we describe electron transport through quantum dots with strong charging effects beyond perturbation theory in the tunneling, and we analyze current fluctuations in a diffusive interacting conductor.Comment: To be published in special issue of "Fortschritte der Physik" (ed. by Wolfgang Schleich

Parton Distribution Function Uncertainties

We present parton distribution functions which include a quantitative estimate of its uncertainties. The parton distribution functions are optimized with respect to deep inelastic proton data, expressing the uncertainties as a density measure over the functional space of parton distribution functions. This leads to a convenient method of propagating the parton distribution function uncertainties to new observables, now expressing the uncertainty as a density in the prediction of the observable. New measurements can easily be included in the optimized sets as added weight functions to the density measure. Using the optimized method nowhere in the analysis compromises have to be made with regard to the treatment of the uncertainties.We present parton distribution functions which include a quantitative estimate of its uncertainties. The parton distribution functions are optimized with respect to deep inelastic proton data, expressing the uncertainties as a density measure over the functional space of parton distribution functions. This leads to a convenient method of propagating the parton distribution function uncertainties to new observables, now expressing the uncertainty as a density in the prediction of the observable. New measurements can easily be included in the optimized sets as added weight functions to the density measure. Using the optimized method nowhere in the analysis compromises have to be made with regard to the treatment of the uncertainties