  When computing quantum-mechanical observables, the ``curse of
dimensionality'' limits the naive approach that uses the quantum-mechanical
wavefunction. The semiclassical Herman--Kluk propagator mitigates this curse by
employing a grid-free ansatz to evaluate the expectation values of these
observables. Here, we investigate quadrature techniques for this
high-dimensional and highly oscillatory propagator. In particular, we analyze
Monte Carlo quadratures using three different initial sampling approaches. The
first two, based either on the Husimi density or its square root, are
independent of the observable whereas the third approach, which is new,
incorporates the observable in the sampling to minimize the variance of the
Monte Carlo integrand at the initial time. We prove sufficient conditions for
the convergence of the Monte Carlo estimators and provide convergence error
estimates. The analytical results are validated by numerical experiments in
various dimensions on a harmonic oscillator and on a Henon-Heiles potential
with an increasing degree of anharmonicity.
