A globally convergent approach for quantum control and system identification*

Numerical optimization methods, collectively referred to as ``optimal control", are an important tool able to determine what is possible in the manipulation of quantum systems when no other method can. Two significant difficulties limit the power of these methods: the need to numerically simulate the evolution of the quantum system many times, and the difficulty in finding near-optimal solutions in the presence of multiple local minima. Here we introduce a formulation of quantum control that removes both of these difficulties. The Magnus expansion and an appropriate representation for the control functions provide an analytic approximation to the evolution of essentially any system that is also polynomial in the control parameters. This approximation both eliminates the need for repeated simulation and allows global optimization with recently-developed optimization methods for polynomials. It greatly reduces the numerical overhead in fixed-time quantum control, minimum-time control, and Hamiltonian identification.

We also show that the problem of obtaining Hamiltonians, Markovian master equations, and Kraus maps for quantum systems from noisy time-series can be reduced to that of polynomial optimization. The importance of this fact is that this special class of optimization problems is essentially solved: sophisticated methods have been devised that will find the global optimum, and gradient search methods can often do so as well. For closed systems the polynomial is quadratic and can be solved directly. For open systems multiple formulations are possible. We illustrate the power of these methods by inferring novel Lindblad master equations from ab into simulations of a qubit coupled to a bosonic bath.