24.01.2017
A. Delgado  "Stochastic optimization on complex variables and tomography of pure states"
Center for Optics and Photonics
Universidad de Concepción, Chile
Organiser: Max Planck Institute for the Science of Light
Time, place: Tuesday, January 24, 2017, 11 a.m. Small Seminar room A.2.500, Staudtstr. 2
Abstract: Realvalued functions of complex arguments violate the CauchyRiemann conditions and, consequently, do not posses a Taylor series expansion on its complex arguments. Therefore, optimiza tion methods based on derivatives cannot be directly applied to this class of functions, which is circumvented by mapping the problem to the field of the real numbers by considering real and imaginary parts of the complex arguments as the new variables. Here, we introduce a stochastic optimization method which works completely within the field of the complex numbers. Thereby, it becomes unnecessary the use of a real parametrization of the complex arguments of the target function. Furthermore, it is possible to exploit the information about the structure of the target function to efficiently solve the optimization problem. The method produces a sequence of estima tions approaching the solution. Each estimation is generated by evaluating the target function at two different randomly chosen points. Fundamental properties of the method, such as asymptotic convergence and unbiasedness, are demonstrated. It also exhibits a large performance enhancement. This is demonstrated by comparing its performance with other algorithms in the case of quantum tomography of pure quantum states. The new method provides solutions which can be two orders of magnitude closer to the true minima or achieve similar results as other methods but with three orders of magnitude less resources.
