Interesting problems in quantum computation take the form of finding low-energy states of (pseudo)spin systems with engineered Hamiltonians that encode the problem data. Motivated by the practical possibility of producing very low-temperature spin sy
stems, we propose and exemplify the possibility to compute by coupling the computational spins to a non-Markovian bath of spins that serve as a heat sink. We demonstrate both analytically and numerically that this strategy can achieve quantum advantage in the Grover search problem.
I show that the cloneability of information is the key difference between classical computer and quantum computer. As information stored and processed by neurons is cloneable, brain (human or non-human) is a classical computer. Penrose argued with th
e Godel theorem that human brain is not classical. I demonstrate with an example why his argument is flawed. At the end, I discuss how to go beyond quantum computer.
We propose a change-point detection method for large scale multiple testing problems with data having clustered signals. Unlike the classic change-point setup, the signals can vary in size within a cluster. The clustering structure on the signals ena
bles us to effectively delineate the boundaries between signal and non-signal segments. New test statistics are proposed for observations from one and/or multiple realizations. Their asymptotic distributions are derived. We also study the associated variance estimation problem. We allow the variances to be heteroscedastic in the multiple realization case, which substantially expands the applicability of the proposed method. Simulation studies demonstrate that the proposed approach has a favorable performance. Our procedure is applied to {an array based Comparative Genomic Hybridization (aCGH)} dataset.
Accurate forecasting is one of the fundamental focus in the literature of econometric time-series. Often practitioners and policy makers want to predict outcomes of an entire time horizon in the future instead of just a single $k$-step ahead predicti
on. These series, apart from their own possible non-linear dependence, are often also influenced by many external predictors. In this paper, we construct prediction intervals of time-aggregated forecasts in a high-dimensional regression setting. Our approach is based on quantiles of residuals obtained by the popular LASSO routine. We allow for general heavy-tailed, long-memory, and nonlinear stationary error process and stochastic predictors. Through a series of systematically arranged consistency results we provide theoretical guarantees of our proposed quantile-based method in all of these scenarios. After validating our approach using simulations we also propose a novel bootstrap based method that can boost the coverage of the theoretical intervals. Finally analyzing the EPEX Spot data, we construct prediction intervals for hourly electricity prices over horizons spanning 17 weeks and contrast them to selected Bayesian and bootstrap interval forecasts.
This paper introduces and analyzes a preconditioned modified of the Hermitian and skew-Hermitian splitting (PMHSS). The large sparse continuous Sylvester equations are solved by PMHSS iterative algorithm based on nonHermitian, complex, positive defin
ite/semidefinite, and symmetric matrices. We prove that the PMHSS is converged under suitable conditions. In addition, we propose an accelerated PMHSS method consisting of two preconditioned matrices and two iteration parameters {alpha}, b{eta}. Theoretical analysis showed that the convergence speed of the accelerated PMHSS is faster compared to the PMHSS. Also, the robustness and efficiency of the proposed two iterative algorithms were demonstrated in numerical experiments.
A general class of time-varying regression models is considered in this paper. We estimate the regression coefficients by using local linear M-estimation. For these estimators, weak Bahadur representations are obtained and are used to construct simul
taneous confidence bands. For practical implementation, we propose a bootstrap based method to circumvent the slow logarithmic convergence of the theoretical simultaneous bands. Our results substantially generalize and unify the treatments for several time-varying regression and auto-regression models. The performance for ARCH and GARCH models is studied in simulations and a few real-life applications of our study are presented through analysis of some popular financial datasets.
We present a quantum algorithm for approximating maximum independent sets of a graph based on quantum non-Abelian adiabatic mixing in the sub-Hilbert space of degenerate ground states, which generates quantum annealing in a secondary Hamiltonian. For
both sparse and dense graphs, our quantum algorithm on average can find an independent set of size very close to $alpha(G)$, which is the size of the maximum independent set of a given graph $G$. Numerical results indicate that an $O(n^2)$ time complexity quantum algorithm is sufficient for finding an independent set of size $(1-epsilon)alpha(G)$. The best classical approximation algorithm can produce in polynomial time an independent set of size about half of $alpha(G)$.
We present an algorithm for the generalized search problem (searching $k$ marked items among $N$ items) based on a continuous Hamiltonian and exploiting resonance. This resonant algorithm has the same time complexity $O(sqrt{N/k})$ as the Grover algo
rithm. A natural extension of the algorithm, incorporating auxiliary monitor qubits, can determine $k$ precisely, if it is unknown. The time complexity of our counting algorithm is $O(sqrt{N})$, similar to the best quantum approximate counting algorithm, or better, given appropriate physical resources.
The monopole for the geometric curvature is studied for non-Hermitian systems. We find that the monopole contains not only the exceptional points but also branch cuts. As the mathematical choice of branch cut in the complex plane is rather arbitrary,
the monopole changes with the branch-cut choice. Despite this branch-cut dependence, our monopole is invariant under the $GL(l,mathbb{C})$ gauge transformation that is inherent in non-Hermitian systems. Although our results are generic, they are presented in the context of a two-mode non-Hermitian Dirac model. A corresponding two-mode Hermitian system is also discussed to illustrate the essential difference between monopoles in Hermitian systems and non-Hermitian systems.
The mean-field dynamics of a Bose gas is shown to break down at time $tau_h = (c_1/gamma) ln N$ where $gamma$ is the Lyapunov exponent of the mean-field theory, $N$ is the number of bosons, and $c_1$ is a system-dependent constant. The breakdown time
$tau_h$ is essentially the Ehrenfest time that characterizes the breakdown of the correspondence between classical and quantum dynamics. This breakdown can be well described by the quantum fidelity defined for reduced density matrices. Our results are obtained with the formalism in particle-number phase space and are illustrated with a triple-well model. The logarithmic quantum-classical correspondence time may be verified experimentally with Bose-Einstein condensates.
