Stochastic optimization on complex variables and pure-state quantum tomography.

Area of Science:

• Complex Analysis
• Optimization Theory
• Quantum Information Science

Background:

• Real-valued functions of complex arguments do not satisfy Cauchy-Riemann conditions, precluding direct application of derivative-based optimization.
• Existing methods often map complex problems to real numbers, potentially complicating analysis and limiting efficiency.
• Optimization of functions with unknown parameters is a significant challenge in various scientific domains.

Purpose of the Study:

• To develop a stochastic optimization method that operates directly in the field of complex numbers.
• To leverage the inherent structure of complex numbers for simpler analysis and improved performance.
• To enable optimization of functions with unknown parameters and demonstrate performance enhancements.

Main Methods:

• Introduced a stochastic optimization algorithm designed to work directly with complex arguments.
• The method generates estimates by evaluating the target function at two randomly selected points.
• Convergence is analyzed asymptotically in the mean towards the optimizer.

Main Results:

• The complex-valued optimization method offers simpler equations and easier analysis compared to real-valued mappings.
• Significant performance enhancements were observed, particularly in quantum tomography of pure states.
• The method achieved solutions orders of magnitude closer to true minima or comparable results with drastically reduced resources.

Conclusions:

• The proposed stochastic optimization method effectively handles real-valued functions of complex arguments by operating within the complex field.
• This approach provides a more efficient and powerful alternative to traditional derivative-based or real-mapped methods.
• The technique shows substantial promise for applications requiring complex number optimization, such as quantum information processing.