Quantum Integration Networks for Efficient Monte Carlo in High-Energy Physics

TL;DR

This work addresses slow convergence and precision limits in high-energy physics Monte Carlo integration by introducing QuInt-Net, a framework of variational quantum circuits that learn antiderivatives $Q(x;\theta)$ so that $I \approx Q(x_{\text{final}};\theta_{\text{opt}}) - Q(x_{\text{init}};\theta_{\text{opt}})$. It investigates data sampling strategies (Uniform, Importance Sampling, and Hamiltonian Monte Carlo) and loss designs (MSE, $\chi^2$, Log-Cosh, and MSE+KL) to enhance integration of functions with singular features. Benchmark studies on Complex Periodic Function, Step Function, and Breit-Wigner distributions show QNN-based QuInt-Net achieving high derivative fidelity ($R^2$ typically > 0.99 for CPF, and >0.98 in many cases) and robust global integral behavior measured by $W_1$, with IS+MSE+KL excelling for oscillatory cases and regularized losses aiding resonance-like integrands. The results indicate feasibility and guidance for applying quantum-integrated Monte Carlo to collider phase-space integrals, while highlighting that performance gains are problem-dependent and that real hardware noise favors certain loss-sampling combinations; extending to higher dimensions and hardware implementations remains a key direction.

Abstract

Monte Carlo methods play a central role in particle physics, where they are indispensable for simulating scattering processes, modeling detector responses, and performing multi-dimensional integrals. However, traditional Monte Carlo methods often suffer from slow convergence and insufficient precision, particularly for functions with singular features such as rapidly varying regions or narrow peaks. Quantum circuits provide a promising alternative: compared to conventional neural networks, they can achieve rich expressivity with fewer parameters, and the parameter-shift rule provides an exact analytic form for circuit gradients, ensuring precise optimization. Motivated by these advantages, we investigate how sampling strategies and loss functions affect integration efficiency within the \textbf{Quantum Integration Network} (QuInt-Net). We compare adaptive and non-adaptive sampling approaches and examine the impact of different loss functions on accuracy and convergence. Furthermore, we explore three quantum circuit architectures for numerical integration: the data re-uploading model, the quantum signal processing protocol, and deterministic quantum computation with one qubit. The results provide new insights into optimizing QuInt-Nets for applications in high energy physics.

Figures (6)

• Figure 1: Schematic illustration of the variational quantum circuit (VQC) workflow for numerical integration. Training data are generated using various sampling strategies, including uniform, importance, and adaptive sampling. The VQC model $Q(x;\theta)$, implemented with QNN, QSP, or DQC1 architectures, is trained to approximate both $f(x)$ and its derivative $q(x)=\partial_x Q(x;\theta)$. A classical optimizer minimizes a loss function such as MSE, $\chi^2$, Log-Cosh, or KL-augmented MSE. The trained VQC model can then be used to calculate the integral of the target function via the optimized parameters.
• Figure 2: Sample distributions obtained from three different sampling strategies: uniform (blue), importance sampling (IS, red), and Hamiltonian Monte Carlo (HMC, green). The target function is shown with black solid lines. Figure in (a) shows results for the Complex Periodic Function (CPF), a highly oscillatory integrand. Figure in (b) shows the Breit–Wigner distribution, where both IS and HMC successfully concentrate samples near the resonance peak, demonstrating their effectiveness in capturing localized features. Each functions are introduced in Section \ref{['sec:CPF']} and Sec \ref{['sec:BW']}.
• Figure 3: QNN simulation results with the CPF example for the (Uniform, MSE) and three configurations with the lowest $W_1$ over the range is $s \in [-2\pi, 2\pi]$ with the normalized variable $s_{\text{norm}}$. (a) derivative fit $q(x;\theta)$ versus target $f(x)$ (bottom strips: pointwise relative error, log scale); (b) sub-interval integrals over $[0,\pi/2]$, $[-\pi/2,\pi/2]$, and $[-\pi,\pi]$; (c) normalized cumulative integral. The curves show the target, the baseline, and the three top configurations. The numerical estimations ($R^2$, $W_1$, and interval errors) are summarized in Table \ref{['tab:CPF_result']}. In panel (c), sharp peaks appear because the reference value is $0$ at $s = 0$ and $2\pi$. At $s = \pi/2$, the (IS, MSE+KL) configuration exhibits particularly high accuracy in reproducing the cumulative integral relative to the reference.
• Figure 4: QNN simulation results with the STEP example for the (Uniform,MSE) and three configurations with the lowest $W_1$ over the range is $s \equiv s_{\text{norm}} \in [-1, 1]$. (a) derivative fit $q(x;\theta)$ versus target $f(x)$ (bottom strips: pointwise relative error, log scale); (b) sub-interval integrals over $[-0.5,0]$, $[0,0.5]$, and $[-0.5,0.5]$; (c) normalized cumulative integral. Curves are composed with the same configurations as in Fig. \ref{['fig:CPF_result']}. Numerical metrics ($R^2$, $W_1$, and interval errors) are summarized in Table \ref{['tab:step_result']} (full QNN results in Table \ref{['tab:qnn_step_result']}). In panel (c), a peak appears at $s = 1$ due to the vanishing reference value, while at $s = 0$ the prediction coincides with the reference as a result of the normalization procedure.
• Figure 5: QNN simulation results with the BW example for the (Uniform, MSE) and three configurations with the lowest $W_1$ over the range $s \in [60, 120]$ with the normalized variable $s_{\text{norm}} \in [-1, 1]$. Panels: (a) derivative fit $q(s;\theta)$ vs. target $f(s)$ (bottom strips: pointwise relative error, log scale); (b) sub-interval integrals over $[M_Z\pm3\Gamma]$, $[M_Z\pm5\Gamma]$, and $[M_Z\pm10\Gamma]$; (c) normalized cumulative integral. Curves is composed with the same configuration as before in Fig. \ref{['fig:CPF_result']}. Numerical metrics ($R^2$, $W_1$, and interval errors) are summarized in Table \ref{['tab:BW_result']} (full QNN results in Table \ref{['tab:qnn_bw_result']}).