Hybrid quantum-classical machine learning potential with variational quantum circuits

Abstract

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.