Predicting features of complex, large-scale quantum systems is essential to the characterization and engineering of quantum architectures. We present an efficient approach for predicting a large number of linear features using classical shadows obtained from very few quantum measurements. This approach is guaranteed to accurately predict @math linear functions with bounded Hilbert-Schmidt norm from only @math measurement repetitions. This sampling rate is completely independent of the system size and saturates fundamental lower bounds from information theory. We support our theoretical findings with numerical experiments over a wide range of problem sizes (2 to 162 qubits). These highlight advantages compared to existing machine learning approaches.
Direct fidelity estimation is a procedure that allows for predicting a single pure target fidelity @math up to accuracy @math . The best-known technique is based on few Pauli measurements that are selected randomly using importance sampling @cite_49 . The required number of samples depends on the target: it can range from a dimension-independent order of @math (if @math is a stablizer state) to roughly @math in the worst case.
@cite_49
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We describe a simple method for certifying that an experimental device prepares a desired quantum state ρ. Our method is applicable to any pure state ρ, and it provides an estimate of the fidelity between ρ and the actual (arbitrary) state in the lab, up to a constant additive error. The method requires measuring only a constant number of Pauli expectation values, selected at random according to an importance-weighting rule. Our method is faster than full tomography by a factor of d, the dimension of the state space, and extends easily and naturally to quantum channels. Abstract Modifications of Prony's classical technique for estimating rate constants in exponential fitting problems have many contemporary applications. In this article the consistency of Prony's method and of related algorithms based on maximum likelihood is discussed as the number of observations n → ∞ by considering the simplest possible models for fitting sums of exponentials to observed data. Two sampling regimes are relevant, corresponding to transient problems and problems of frequency estimation, each of which is associated with rather different kinds of behavior. The general pattern is that the stronger results are obtained for the frequency estimation problem. However, the algorithms considered are all scaling dependent and consistency is not automatic. A new feature that emerges is the importance of an appropriate choice of scale in order to ensure consistency of the estimates in certain cases. The tentative conclusion is that algorithms referred to as Objective function Reweighting Algorithms ... We propose a new method for pointwise estimation of monotone, uni- modal and U-shaped failure rates, under a right-censoring mechanism, using non- parametric likelihood ratios. The asymptotic distribution of the likelihood ratio is pivotal, though non-standard, and can therefore be used to construct asymptotic confidence intervals for the failure rate at a point of interest, via inversion. Major advantages of the new method lie in the facts that it completely avoids estima- tion of nuisance parameters, or the choice of a bandwidth tuning parameter, and is extremely easy to implement. The new method is shown to perform competi- tively in simulations, and is illustrated on a data set involving time to diagnosis of schizophrenia in the Jerusalem Perinatal Cohort Schizophrenia Study. Direct imaging technology is an effective and convenient method for the estimation of bubble size distribution (BSD). However, overlapping bubble has an influence on BSD when gas holdup is more than 1 . In this paper, we present a new method of overlapping elliptical bubble recognition to determine bubble size. The method mainly includes two steps: contour segmentation and segment grouping. Contour segmentation is on the assumption that the concave points in the dominant point sequence are always the connecting points, and segment grouping is mainly based on the average distance deviation criterion. Both simulated images and real bubble images are used to evaluate this new method. The results show that it is effective in the recognition of overlapping elliptical bubbles and have a potential in other elliptical object recognition. In the last, two methods are used for BSD estimation. It is found that the bubble size (such d10 or d32) estimated by the ignore method is slightly smaller than that estimated by the recognition method.