Quantum Speedup in Molecular Integral Evaluation through Angular Momentum Coupling
Hang Hu, Gilles H. Peslherbe, Hsu Kiang Ooi, Anguang Hu · IntechOpen eBooks · 2025
This chapter introduces a novel computational formalism utilizing Solid Harmonic Gaussian Orbitals (SHGOs) for efficiently calculating molecular integrals. Despite the prevalent utilization of Cartesian Gaussian Orbitals (CGOs) attributed to their straightforward nature, they inadequately capitalize on the intrinsic rotational symmetry characteristic of quantum systems. SHGOs, in contrast, clearly separate radial and angular components and align naturally with angular momentum symmetries. This alignment significantly enhances computational efficiency and accuracy in quantum chemical calculations. The approach described here leverages advanced angular momentum algebra techniques, particularly vector-coupling and vector-uncoupling schemes, to systematically reduce the complexity of molecular integral evaluations. By leveraging Clebsch-Gordan coefficients and Wigner 3-j symbols, the developed formalism significantly enhances computational efficiency over conventional CGO methods, as evidenced by a notable decrease in floating-point operations. Moreover, this SHGO-based method offers promising connections to emerging quantum computing applications. Quantum computing architectures inherently employ angular momentum entanglement principles, rendering SHGOs exceptionally well-suited for native quantum simulations. Specifically, angular momentum entanglement networks constructed from Clebsch-Gordan transformations capture rotational properties effectively, enabling scalable quantum simulation platforms and enhancing computational accuracy. Additionally, this methodology has significant implications beyond traditional quantum chemistry, offering pathways for quantum sensing and imaging technologies through the explicit consideration of orbital angular momentum. This integrated framework thus positions SHGO-based methods at the forefront of computational quantum chemistry, providing a critical tool to bridge classical methodologies with quantum computational advancements.