Stochastic optimization on complex variables and pure-state quantum tomography

Abstract

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.

Estimation of pure quantum states in high dimension at the limit of quantum accuracy through complex optimization and statistical inference

“…For the same states CSPSA without MLE yields after 10 iterations a mean infidelity approximately equal to and (see Fig. 1 in 30 ), for increasing . Thus, the concatenation of CSPSA to MLE yields a mean infidelity that is to times closer to the true minimum than the one provided by CSPSA alone.…”

“…Thereby, the complex gradient is defined by with . The CSPSA method is defined by the iterative rule 30 where is a positive gain coefficient and is the estimate of the minimizer of at the k-th iteration. The iteration starts from an initial guess , which is randomly chosen.…”

“…SPSA has been proposed 42 and experimentally demonstrated 44 as tomographic method for pure quantum states. In this context, it has been show 30 that CSPSA achieves a higher convergence rate than SPSA. …”

“…In the case of pure states the quantum accuracy is limited by the Gill-Massar lower bound 29 . The method is based on the concatenation of the Complex simultaneous perturbation stochastic approximation (CSPSA), a recently proposed iterative stochastic optimization method on the field of the complex numbers 30 , and Maximum likelihood estimation (MLE), a well known statistical inference method 31 . The method here proposed reaches the quantum accuracy limit after a small number of iterations, typically of the order of 8, for all inspected dimensions.…”

Self-Guided Quantum State Learning for Mixed States

“…Some recent proposals for quantum state tomography include self-guided quantum tomography (SGQT), practical adaptive quantum tomography (PAQT), and state tomography through eigenstate extraction with neural networks [17][18][19][20][21][22]. SGQT employs a stochastic approximation optimization technique known as simultaneous perturbation stochastic approximation (SPSA) [23] to learn an unknown pure state.…”