Hybrid quantum-classical machine learning potential with variational quantum circuits

Machine learning potentials (MLPs) based on equivariant message-passing neural networks have achieved state-of-the-art accuracy for atomistic simulation, yet integration of variational quantum circuit (VQC) readouts into such frameworks remains unexplored for condensed-matter systems. Here we employ and characterize a hybrid quantum-classical MLP (HQC-MLP) by coupling an E(3)-equivariant message-passing network with VQC readout layers for predicting DFT energies and forces of liquid silicon (64 atoms, periodic cell, 2000-3000 K). We benchmark five VQC Ansatz architectures differing in entangling gate type (controlled-Z (CZ) vs. controlled-X (CX)) and circuit structure against a classical baseline. All HQC-MLP variants reproduce PBE-DFT energies with test-set RMSEs of ∼3.5 meV per atom when evaluated using exact expectation values, while entangled circuits exhibit faster training convergence in epoch count, although not in wall-clock time. CZ-based entangling gates achieve lower validation losses than their CX-based counterparts. This trend is consistent with two factors: the phase-preserving nature of CZ gates for amplitude-encoded features and their reduced effective entangling depth, since the final CZ layer commutes with the Pauli-Z measurement operators. Both molecular dynamics and Monte Carlo simulations accurately reproduce the DFT radial distribution function at 2000 K, despite the models being trained exclusively on 3000 K configurations; molecular dynamics also reproduces the Si-Si-Si bond-angle distribution. Finite-shot analysis further shows that achieving an RMSE of approximately 2 meV per atom relative to shot-noise-free expectation values requires about 5 × 10⁴ shots per circuit execution. This corresponds to approximately 2.56 × 10⁸ shots for a single configuration force evaluation, making direct force calculations prohibitively expensive on current noisy intermediate-scale quantum (NISQ) hardware and thereby motivating the proposed Monte Carlo protocol. Finally, the cutoff-radius sensitivity analysis reveals that HQC-MLP exhibits truncation behavior comparable to that of the classical MLP.