Limitations of Average Hamiltonian Theory for Quantum Control

One path to Hamiltonian Engineering in solid-state spin systems is to use periodic pulse trains to make the system evolve under a desired effective Hamiltonian. These pulse trains are designed using Average Hamiltonian Theory, which uses the Magnus series expansion to obtain a time-independent description of a periodic time-dependent Hamiltonian at stroboscopic intervals. The most common application of these pulse trains has been to decouple internuclear magnetic dipolar interactions in solid state NMR. These sequences have been improved by increasing their length and complexity in order to cancel out higher order terms in this expansion. It has generally been believed that the Magnus expansion converges rapidly in the parameter regime in which these sequences are used, and that therefore the largest source of error in implementing these sequences comes from the lowest terms in the expansion that the pulse sequence does not account for. In an effort to explore the limits of control, we numerically explore if the maximum attainable fidelity of conventional AHT-designed dipolar decoupling sequences is due to their sensitivity to experimental error or due to poor convergence of the Magnus series.