Engineering effective Hamiltonians

Abstract

In the field of quantum control, effective Hamiltonian engineering is a powerful tool that utilizes perturbation theory to mitigate or enhance the effect that a variation in the Hamiltonian has on the evolution of the system. Here, we provide a general framework for computing arbitrary timedependent perturbation theory terms, as well as their gradients with respect to control variations, enabling the use of gradient methods for optimizing these terms. In particular, we show that effective Hamiltonian engineering is an instance of a bilinear control problem-the same general problem class as that of standard unitary design-and hence the same optimization algorithms apply. We demonstrate this method in various examples, including decoupling, recoupling, and robustness to control errors and stochastic errors. We also present a control engineering example that was used in experiment, demonstrating the practical feasibility of this approach.

Hybrid Quantum-Classical Approach to Quantum Optimal Control

“…As observed by Ref. [28], since the Van Loan equation has the same general form as the Schrödinger equation, which implies that Eq. (7) still belongs to the class of bilinear control theory problems [33], thus evaluation of ∂Φ/∂u[m] can be done in the same way as how gradients are estimated in GRAPE.…”

“…Furthermore, I show that the robust control problem can be effectively solved by a modified GRAPE algorithm under the Van Loan differential equation framework that was recently developed in Ref. [28].It is worth pointing out that the feasibility of the approach crucially relies on assuming that the inter-subsystem couplings can be treated as perturbation terms. In principle, the validity of the assumption is relevant to how the full system is divided.…”

“…Furthermore, I show that the robust control problem can be effectively solved by a modified GRAPE algorithm under the Van Loan differential equation framework that was recently developed in Ref. [28].…”

“…Here, the first line can be obtained via the Dyson series [28,29]. Therefore, f S k Sj can be evaluated simply on the pair of subsystems S k and S j .…”

“…To this end, I follow the methodology developed by Ref. [28], which extends the Shrödinger equation to a socalled Van Loan block matrix differential equation by which means the same gradient-based algorithm can apply. Concretely, define a set of operators L S k Sj : 1 ≤ k < j ≤ s and {L C k (t) : k = 1, .…”