Stochastic optimization on complex variables and pure-state quantum tomography

Real-valued functions of complex arguments violate the Cauchy-Riemann conditions and, consequently, do not have Taylor series expansion. Therefore, optimization methods based on derivatives cannot be directly applied to this class of functions. This 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 independent variables. We introduce a stochastic optimization method that works within the field of the complex numbers. This has two advantages: Equations on complex arguments are simpler and easy to analyze and the use of the complex structure leads to performance improvements. The method produces a sequence of estimates that converges asymptotically in mean to the optimizer. Each estimate is generated by evaluating the target function at two different randomly chosen points. Thereby, the method allows the optimization of functions with unknown parameters. Furthermore, the method exhibits a large performance enhancement. This is demonstrated by comparing its performance with other algorithms in the case of quantum tomography of pure states. The 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.