SpinGQE: A Generative Quantum Eigensolver for Spin Hamiltonians

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 (9)

• 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 $J=h=10$. The plot shows the energy achieved by generated circuits as a function of training epoch. Model size is 37M parameters, with hyperparameters $\beta = 0.3$ and $M=10$ circuits per epoch. The shaded regions correspond to the range of energies of these $10$ 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 $J=1, h=10$. The field-dominated regime exhibits smoother convergence to the ground state compared to the antiferromagnetic regime, reflecting the simpler energy landscape.