Upper bounds on quantum query complexity inspired by the Elitzur-Vaidman bomb tester

Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002], we introduce a new query complexity model, which we call bomb query complexity $B(f)$. We investigate its relationship with the usual quantum query complexity $Q(f)$, and show that $B(f)=\Theta(Q(f)^2)$. This result gives a new method to upper bound the quantum query complexity: we give a method of finding bomb query algorithms from classical algorithms, which then provide nonconstructive upper bounds on $Q(f)=\Theta(\sqrt{B(f)})$. We subsequently were able to give explicit quantum algorithms matching our upper bound method. We apply this method on the single-source shortest paths problem on unweighted graphs, obtaining an algorithm with $O(n^{1.5})$ quantum query complexity, improving the best known algorithm of $O(n^{1.5}\sqrt{\log n})$ [arXiv:quant-ph/0606127]. Applying this method to the maximum bipartite matching problem gives an $O(n^{1.75})$ algorithm, improving the best known trivial $O(n^2)$ upper bound.Comment: 32 pages. Minor revisions and corrections. Regev and Schiff's proof that P(OR) = \Omega(N) remove

Nonparametric modeling and forecasting electricity demand: an empirical study

This paper uses half-hourly electricity demand data in South Australia as an empirical study of nonparametric modeling and forecasting methods for prediction from half-hour ahead to one year ahead. A notable feature of the univariate time series of electricity demand is the presence of both intraweek and intraday seasonalities. An intraday seasonal cycle is apparent from the similarity of the demand from one day to the next, and an intraweek seasonal cycle is evident from comparing the demand on the corresponding day of adjacent weeks. There is a strong appeal in using forecasting methods that are able to capture both seasonalities. In this paper, the forecasting methods slice a seasonal univariate time series into a time series of curves. The forecasting methods reduce the dimensionality by applying functional principal component analysis to the observed data, and then utilize an univariate time series forecasting method and functional principal component regression techniques. When data points in the most recent curve are sequentially observed, updating methods can improve the point and interval forecast accuracy. We also revisit a nonparametric approach to construct prediction intervals of updated forecasts, and evaluate the interval forecast accuracy.Functional principal component analysis; functional time series; multivariate time series, ordinary least squares, penalized least squares; ridge regression; seasonal time series