Universal 2-Local Symmetry-Preserving Quantum Neural Networks for Fermionic Systems
Simulating quantum many-body systems represents a fundamental challenge where classical machine learning methods are severely bottlenecked by the exponential curse of dimensionality. Variational Quantum Algorithms (VQAs) offer a native paradigm to tackle this by optimizing parameterized unitary evolutions to find the ground states of problem Hamiltonians. However, the efficacy of these VQA is deeply hindered by the challenge of balancing the preservation of critical physical symmetries with the strict constraints of hardware implementability. In this work, we address this dilemma by proposing a hardware-efficient, symmetry-preserving ansatz fortified with complete theoretical guarantees for fermionic systems, termed the Hamming Weight Preserving (HWP) ansatz. We establish the necessary and sufficient conditions for 2-local HWP operators to achieve subspace universality, formally debunking the prevailing assumption that truncation-free simulation requires complex high-order interactions. Empirical validations corroborate our theoretical guarantees, showcasing the exact approximation of arbitrary unitary matrices within the HWP subspace. Crucially, we demonstrate the exceptional versatility of the proposed approach by deploying the exact same ansatz across distinct fermionic models, including diverse molecular electronic structures and the Fermi-Hubbard model. Our proposed HWP ansatz consistently suppresses ground-state energy errors below $1 \times 10^{-10}$ Ha, achieving a level of precision that surpasses the stringent threshold of chemical accuracy by multiple orders of magnitude. This work establishes a complete, theoretically fortified 2-local framework for symmetry-preserving computation, offering a highly universal and hardware-efficient building block for advancing quantum machine learning and fermionic many-body simulations.