  We study the time evolution of mean values of quantum operators in a regime
plagued by two difficulties: The smallness of $\hbar$ and the presence of
strong and ubiquitous classical chaos. While numerics become too
computationally expensive for purely quantum calculations as $\hbar \to 0$,
methods that take advantage of the smallness of $\hbar$ -- that is,
semiclassical methods -- suffer from both conceptual and practical difficulties
in the deep chaotic regime. We implement an approach which addresses these
conceptual problems, leading to a deeper understanding of the origin of the
interference contributions to the operator's mean value. We show that in the
deep chaotic regime our approach is capable of unprecedented accuracy, while a
standard semiclassical method (the Herman-Kluk propagator) produces only
numerical noise. Our work paves the way to the development and employment of
more efficient and accurate methods for quantum simulations of systems with
strongly chaotic classical limits.
