# Engineering Effective Hamiltonians

## TL;DR

This paper introduces a comprehensive framework for designing effective Hamiltonians in quantum control, enabling optimized manipulation of quantum systems through gradient-based methods and demonstrating practical experimental applications.

## Contribution

It provides a general method for computing time-dependent perturbation terms and their gradients, framing effective Hamiltonian engineering as a bilinear control problem with broad applicability.

## Key findings

- Framework for arbitrary time-dependent perturbation calculations
- Application of gradient methods for Hamiltonian optimization
- Experimental demonstration of control engineering in quantum systems

## Abstract

In the field of quantum control, effective Hamiltonian engineering is a powerful tool that utilises 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 time-dependent 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.