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Sampling strategies for expectation values within the Herman--Kluk approximation

Fabian Kröninger, Caroline Lasser, Jiri J. L. Vanicek

TL;DR

The paper tackles the challenge of computing quantum observables in high-dimensional systems by leveraging the Herman–Kluk semiclassical propagator, which recasts expectation values as a double phase-space integral amenable to grid-free Monte Carlo methods. It compares three sampling strategies—Husimi, sqrt-Husimi, and an observable-informed density ρ_opt—analyzing convergence and variance, and introduces a self-normalized weighted estimator (A_N^{W}) that achieves linear scaling when combined with Hamiltonian Monte Carlo. Theoretical results establish conditions for convergence, variance behavior, and MSE scaling, while numerical experiments on harmonic and Henon–Heiles potentials validate the variance-reduction benefits and show substantial efficiency gains, especially in higher dimensions. The study provides practical guidelines for choosing sampling densities and weights, enabling more efficient HK-based simulations and offering pathways to extend these ideas to time-correlation functions and other Gaussian-based quantum methods. Overall, the work delivers a scalable, variance-controlled framework for computing expectation values in semiclassical quantum dynamics.

Abstract

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.

Sampling strategies for expectation values within the Herman--Kluk approximation

TL;DR

The paper tackles the challenge of computing quantum observables in high-dimensional systems by leveraging the Herman–Kluk semiclassical propagator, which recasts expectation values as a double phase-space integral amenable to grid-free Monte Carlo methods. It compares three sampling strategies—Husimi, sqrt-Husimi, and an observable-informed density ρ_opt—analyzing convergence and variance, and introduces a self-normalized weighted estimator (A_N^{W}) that achieves linear scaling when combined with Hamiltonian Monte Carlo. Theoretical results establish conditions for convergence, variance behavior, and MSE scaling, while numerical experiments on harmonic and Henon–Heiles potentials validate the variance-reduction benefits and show substantial efficiency gains, especially in higher dimensions. The study provides practical guidelines for choosing sampling densities and weights, enabling more efficient HK-based simulations and offering pathways to extend these ideas to time-correlation functions and other Gaussian-based quantum methods. Overall, the work delivers a scalable, variance-controlled framework for computing expectation values in semiclassical quantum dynamics.

Abstract

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.
Paper Structure (30 sections, 8 theorems, 133 equations, 6 figures, 2 algorithms)

This paper contains 30 sections, 8 theorems, 133 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Previous research
  3. Main results
  4. Outline of the paper
  5. Notation
  6. The Herman--Kluk propagator
  7. Integral representation
  8. The representation for expectation values
  9. Computational tasks
  10. Probabilistic space discretization
  11. Husimi and sqrt-Husimi sampling
  12. Incorporating the operator and a variational problem
  13. Weighted Importance sampling
  14. Proof of \ref{['Theorem:Properties_WIS']}
  15. Choice of the weight $W$
  16. ...and 15 more sections

Key Result

Lemma 1

Let $\tau>0$ be a fixed time and let $\hat{A}=\mathrm{Id}$, $\hat{q}^n_j$, $\hat{p}^n_j$, or $\hat{H}$ for $n \in \mathbb{N}$, $j \in \{1, \dots , D\}$.

Figures (6)

  • Figure 1: Initial sampling error for $\hat{A}=\mathrm{Id}$ and $\hat{A}=\hat{q}_1$ in one and six dimensions as a function of the number $N$ of Monte Carlo quadrature points. Each panel displays the error for different choices of the weight $W$ as well as the analytical error estimation for $W=\rho_{\rm H}^{\rm dbl}/\rho_{\textup{sqrt-H}}^{\rm dbl}$ and $\hat{A}=\mathrm{Id}$ derived in \ref{['EX:3.6']}.
  • Figure 2: Sampling error of $\hat{A}=\mathrm{Id}$, $\hat{A}=\hat{q}_1$ and $\hat{A}=\hat{p}_1$ in one and ten dimensions as a function of the number $N$ of Monte Carlo points as well as a function of dimension $D$ for a fixed number of samples $N= 2^{20}$. Each panel displays the error for the Husimi (solid line), sqrt-Husimi (dashed line), and optimal approach (dash-dotted line). Theoretical error estimations for the sqrt-Husimi and optimal approaches are displayed with marked lines.
  • Figure 3: Time dependence of the sampling error of the Herman--Kluk expectation values of norm squared, position, momentum and energies propagated in a harmonic oscillator. Each panel is produced by 100 independent simulations each consisting of $N=2^{20}$ trajectories.
  • Figure 4: Time dependence of the averaged Herman--Kluk prefactor in a 6-dimensional Henon-Heiles potential for the three different sampling approaches and three different choices of the initial position $q_0$. The averaged Herman--Kluk prefactors were calculated with $N=2^{17}$ quadrature points each. For each initial condition, we display the total energy $E_{\mathrm{tot}}$.
  • Figure 5: Time evolution of the intrinsic error \ref{['EQ:Intrinsic_error']} using $N=2^{19}\approx 5\times 10^5$ quadrature points for the Husimi (solid line), square root Husimi (triangular marker) and optimal (circular marker) approaches and for three different choices of the initial position $q_0$. Each line was produced by averaging over eight independent simulations.
  • ...and 1 more figures

Theorems & Definitions (47)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Definition 1: Monte Carlo Estimator
  • Lemma 3
  • proof
  • Definition 2: Husimi and sqrt-Husimi densities, Lasser_Lubich:2020
  • ...and 37 more