Quantum Robust Control using Geometric Optimal Control Theory

TL;DR

This paper develops a geometric optimal control framework for quantum robust control, focusing on minimizing sensitivity to model deviations while considering control energy, with explicit solutions for single and two-qubit systems.

Contribution

It introduces a novel geometric optimal control approach to quantum robustness, providing explicit solutions and extending to multi-qubit systems.

Findings

Explicit solutions for single-qubit control minimizing sensitivity and energy.

Extension of the approach to two-qubit systems reducing cross-talk.

The control solutions are smooth and avoid discontinuities.

Abstract

In this paper, we demonstrate an approach to quantum robust control based on the tools of geometric optimal control. The central objects of interest are the sensitivity functions defined as the coefficients in the Taylor expansion of the trajectory with respect to the (unknown, small) parameters which describe the deviation of the actual model from nominal one. In terms of these quantities, we formalize an optimal control problem where one searches for the optimal nominal trajectory which minimizes the size of the sensitivity while taking into account other aspects of the control design such as the energy of the control field. We consider in detail the case of a single qubit with a dephasing Hamiltonian term, and the optimal control problem of obtaining a state transfer by minimizing the weighted sum of the energy of the controlling field and the first order sensitivity. At the limit of a very large weight on the sensitivity, we obtain the optimal control which zeros the sensitivity and minimizes the control field energy. This problem has a rich mathematical structure which enables its solution in terms of elliptic integrals. For this problem, we obtain an explicit solution which is particularly simple and also smooth, avoiding discontinuities which are present in other approaches. We extend the results to the robust control of two quantum bits minimizing cross-talk contamination, as we show that such a problem decouples in two independent one qubit problems.