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.

翻译：在计算量子力学可观测量时，维数灾难限制了直接使用量子力学波函数的朴素方法。半经典的 Herman-Kluk 传播子通过采用无网格拟设来评估这些观测量的期望值，从而缓解了这一灾难。本文研究了针对这一高维且高度振荡的传播子的数值积分技术。特别地，我们分析了使用三种不同初始采样方法的蒙特卡洛积分。前两种方法分别基于 Husimi 密度或其平方根，它们与观测量无关；而第三种方法是新提出的，它将观测量纳入采样过程，以最小化初始时刻蒙特卡洛被积函数的方差。我们证明了蒙特卡洛估计量收敛的充分条件，并给出了收敛误差估计。通过在不同维度下对谐振子以及具有递增非简谐度的 Henon-Heiles 势进行数值实验，验证了理论分析结果。