Hyperbolic Concentration, Anti-concentration, and Discrepancy

Chernoff bound is a fundamental tool in theoretical computer science. It has been extensively used in randomized algorithm design and stochastic type analysis. Discrepancy theory, which deals with finding a bi-coloring of a set system such that the coloring of each set is balanced, has a huge number of applications in approximation algorithms design. Chernoff bound [Che52] implies that a random bi-coloring of any set system with $n$ sets and $n$ elements will have discrepancy $O(\sqrt{n \log n})$ with high probability, while the famous result by Spencer [Spe85] shows that there exists an $O(\sqrt{n})$ discrepancy solution. The study of hyperbolic polynomials dates back to the early 20th century when used to solve PDEs by G{\aa}rding [G{\aa}r59]. In recent years, more applications are found in control theory, optimization, real algebraic geometry, and so on. In particular, the breakthrough result by Marcus, Spielman, and Srivastava [MSS15] uses the theory of hyperbolic polynomials to prove the Kadison-Singer conjecture [KS59], which is closely related to discrepancy theory. In this paper, we present a list of new results for hyperbolic polynomials: * We show two nearly optimal hyperbolic Chernoff bounds: one for Rademacher sum of arbitrary vectors and another for random vectors in the hyperbolic cone. * We show a hyperbolic anti-concentration bound. * We generalize the hyperbolic Kadison-Singer theorem [Br\"a18] for vectors in sub-isotropic position, and prove a hyperbolic Spencer theorem for any constant hyperbolic rank vectors. The classical matrix Chernoff and discrepancy results are based on determinant polynomial. To the best of our knowledge, this paper is the first work that shows either concentration or anti-concentration results for hyperbolic polynomials. We hope our findings provide more insights into hyperbolic and discrepancy theories

Evaluation of Wind Turbine Operation Status Based on ACO + FAHP

Aiming at the shortcomings of the fuzzy analytic hierarchy process (FAHP) in the comprehensive evaluation of wind power projects, such as the diffi culty of satisfying and modifying the consistency of the judgment matrix and the high computational complexity, a fuzzy analytic hierarchy process based on ant colony optimization (ACO+FAHP) is proposed. Firstly, the proposed fuzzy analytic hierarchy process based on ant colony optimization algorithm overcomes the disadvantages that the weight and consistency cannot be improved once the judgment matrix is given. The comparison chart of the consistency ratio calculated according to this method shows that the consistency ratio B, C1-C5 all have diff erent degrees of reduction. Then, in view of the fact that various qualitative indicators cannot be accurately calculated, the wind turbine operating status evaluation model is established by using the fuzzy comprehensive evaluation method. In this paper, the evaluation score of a certain wind farm is 0.731, which means that the operators need to carry out high-level maintenance at this time

QED driven QAOA for network-flow optimization

We present a general framework for modifying quantum approximate optimization algorithms (QAOA) to solve constrained network flow problems. By exploiting an analogy between flow constraints and Gauss's law for electromagnetism, we design lattice quantum electrodynamics (QED) inspired mixing Hamiltonians that preserve flow constraints throughout the QAOA process. This results in an exponential reduction in the size of the configuration space that needs to be explored, which we show through numerical simulations, yields higher quality approximate solutions compared to the original QAOA routine. We outline a specific implementation for edge-disjoint path (EDP) problems related to traffic congestion minimization, numerically analyze the effect of initial state choice, and explore trade-offs between circuit complexity and qubit resources via a particle-vortex duality mapping. Comparing the effect of initial states reveals that starting with an ergodic (unbiased) superposition of solutions yields better performance than beginning with the mixer ground-state, suggesting a departure from the "short-cut to adiabaticity" mechanism often used to motivate QAOA.Comment: 14 pages, 10 figure

Solid Solution Strengthened Fe Alloys

Iron (Fe)-based alloys (such as steel) are widely used structural materials in industry. Numerous methods have been applied to improve their mechanical properties. In this study, we used a technique know as magnetron sputtering to deposit various Fe-based binary alloy coatings to investigate the influence of solutes on solid solution hardening. Several factors contribute to the solid solution hardening of the alloys, such as composition, atomic radius, modulus, and lattice parameter. After preliminary calculations and analysis, we selected several solutes, including molybdenum (Mo), niobium (Nb), and zirconium (Zr). The compositions of solutes were varied to be 2.5, 5, 8 atomic %. Our nanoindentation hardness measurements show that among the three solid solution alloys, Fe-Zr has the highest hardness. The influences of solutes on microstructural and hardness evolution in these solid solution alloys are discussed

Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

Significant effort in applied quantum computing has been devoted to the problem of ground state energy estimation for molecules and materials. Yet, for many applications of practical value, additional properties of the ground state must be estimated. These include Green's functions used to compute electron transport in materials and the one-particle reduced density matrices used to compute electric dipoles of molecules. In this paper, we propose a quantum-classical hybrid algorithm to efficiently estimate such ground state properties with high accuracy using low-depth quantum circuits. We provide an analysis of various costs (circuit repetitions, maximal evolution time, and expected total runtime) as a function of target accuracy, spectral gap, and initial ground state overlap. This algorithm suggests a concrete approach to using early fault tolerant quantum computers for carrying out industry-relevant molecular and materials calculations