Improving non-linear fits

In this notes we describe an algorithm for non-linear fitting which incorporates some of the features of linear least squares into a general minimum $\chi^2$ fit and provide a pure Python implementation of the algorithm. It consists of the variable projection method (varpro), combined with a Newton optimizer and stabilized using the steepest descent with an adaptative step. The algorithm includes a term to account for Bayesian priors. We performed tests of the algorithm using simulated data. This method is suitable, for example, for fitting with sums of exponentials as often needed in Lattice Quantum Chromodynamics

Sequential nonideal measurements of quantum oscillators: Statistical characterization with and without environmental coupling

A one-dimensional quantum oscillator is monitored by taking repeated position measurements. As a first con- tribution, it is shown that, under a quantum nondemolition measurement scheme applied to a system initially at the ground state, (i) the observed sequence of measurements (quantum tracks) corresponding to a single experiment converges to a limit point, and that (ii) the limit point is random over the ensemble of the experiments, being distributed as a zero-mean Gaussian random variable with a variance at most equal to the ground-state variance. As a second contribution, the richer scenario where the oscillator is coupled with a frozen (i.e., at the ground state) ensemble of independent quantum oscillators is considered. A sharply different behavior emerges: under the same measurement scheme, here we observe that the measurement sequences are essentially divergent. Such a rigorous statistical analysis of the sequential measurement process might be useful for characterizing the main quantities that are currently used for inference, manipulation, and monitoring of many quantum systems. Several interesting properties of the quantum tracks evolution, as well as of the associated (quantum) threshold crossing times, are discussed and the dependence upon the main system parameters (e.g., the choice of the measurement sampling time, the degree of interaction with the environment, the measurement device accuracy) is elucidated. At a more fundamental level, it is seen that, as an application of basic quantum mechanics principles, a sharp difference exists between the intrinsic randomness unavoidably present in any quantum system, and the extrinsic randomness arising from the environmental coupling, i.e., the randomness induced by an external source of disturbance.Comment: pages 16 Figures

Probabilistic data flow analysis: a linear equational approach

Speculative optimisation relies on the estimation of the probabilities that certain properties of the control flow are fulfilled. Concrete or estimated branch probabilities can be used for searching and constructing advantageous speculative and bookkeeping transformations. We present a probabilistic extension of the classical equational approach to data-flow analysis that can be used to this purpose. More precisely, we show how the probabilistic information introduced in a control flow graph by branch prediction can be used to extract a system of linear equations from a program and present a method for calculating correct (numerical) solutions.Comment: In Proceedings GandALF 2013, arXiv:1307.416