quant-phMar 25, 2026

SpinGQE: A Generative Quantum Eigensolver for Spin Hamiltonians

Authors: Alexander Holden, Moinul Hossain Rahat, Nii Osae Osae Dade

Organizations: Mindbeam AI† · Department of Computer and Data Sciences, Case Western Reserve University

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Abstract

The ground state search problem is central to quantum computing, with applications spanning quantum chemistry, condensed matter physics, and optimization. The Variational Quantum Eigensolver (VQE) has shown promise for small systems but faces significant limitations. These include barren plateaus, restricted ansatz expressivity, and reliance on domain-specific structure. We present SpinGQE, an extension of the Generative Quantum Eigensolver (GQE) framework to spin Hamiltonians. Our approach reframes circuit design as a generative modeling task. We employ a transformer-based decoder to learn distributions over quantum circuits that produce low-energy states. Training is guided by a weighted mean-squared error loss between model logits and circuit energies evaluated at each gate subsequence. We validate our method on the four-qubit Heisenberg model, demonstrating successfulconvergencetonear-groundstates. Throughsystematichyperparameterexploration, we identify optimal configurations: smaller model architectures (12 layers, 8 attention heads), longer sequence lengths (12 gates), and carefully chosen operator pools yield the most reliable convergence. Our results show that generative approaches can effectively navigate complex energy landscapes without relying on problem-specific symmetries or structure. This provides a scalable alternative to traditional variational methods for general quantum systems. An open-source implementation is available at https://github.com/Mindbeam-AI/SpinGQE.

Figures & tables

Figure 1: SpinGQE methodology for Hamiltonian ground state search. The framework iteratively samples quantum circuits from a transformer model, evaluates their energies on a quantum device, and updates the model to favor low-energy circuits through weighted loss optimization. An optimization loop can further refine the circuits to lower the energy.
Figure 2: Training convergence of SpinGQE on the Heisenberg Hamiltonian with J=h=10J=h=10 . The plot shows the energy achieved by generated circuits as a function of training epoch. Model size is 37M parameters, with hyperparameters β=0.3\beta=0.3 and M=10M=10 circuits per epoch. The shaded regions correspond to the range of energies of these 1010 circuits, while the solid line represents their average.
Figure 3: Learned gate and angle distributions from the final model. (a) Frequency of gate types and qubit pair combinations across 100 sampled circuits. (b) Distribution of rotation angles for each gate type and qubit pair, showing the model’s learned preferences for specific parameter values.
Figure 4: Progressive optimization of a generated circuit through post-processing stages. The base model output (left) undergoes angle refinement (middle) and wire swap optimization (right), achieving successive energy reductions to reach near-exact ground state energy. Circuit diagrams show the gate sequences at each stage.
Figure 5: Training convergence of SpinGQE on the Heisenberg model with J=1,h=10J=1,h=10 . The field-dominated regime exhibits smoother convergence to the ground state compared to the antiferromagnetic regime, reflecting the simpler energy landscape.
Figure 6: Ground state energy estimation across different physical regimes of the Heisenberg model. The SpinGQE method with the optimization loop successfully identifies near-ground states for both antiferromagnetic ( h/J≤1h/J\leq 1 ) and field-dominated ( h/J>1h/J>1 ) regimes, demonstrating robustness across different energy landscape topologies.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Training convergence of the 113M parameter model on the Heisenberg model with J=h=10J=h=10 . The larger model fails to converge reliably due to insufficient training data, highlighting the importance of matching model capacity to available quantum evaluations.
Figure 8: Training convergence with an enlarged operator pool on the Heisenberg model with J=h=10J=h=10 . The expanded operator pool leads to less reliable convergence, demonstrating that a carefully constrained search space improves training efficiency for models with limited data.
Figure 9: Ground state energy of the 1D Heisenberg Hamiltonian ( h=J=10h=J=10 ) as a function of the energy weighting parameter β\beta and circuits per epoch MM , obtained from the 37M parameter model before applying post-processing optimization. The heatmap demonstrates that moderate values of β\beta and relatively small values of MM yield optimal performance.
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