Hamilton-Zero: Foundation Model Computes Quantum Ground States for Up to 8100 Qubits

Researchers presented Hamilton-Zero on August 12, 2026, a Neural Tensor-Network Foundation Model that computes ground states of quadratic qubit Hamiltonians. The model with approximately 0.5 billion parameters was trained on hundreds of thousands of Hamiltonian systems and generalizes from 64 qubits during training to up to 8100 qubits during evaluation – a 127-fold system enlargement.

The architecture formulates the quantum ground state problem as manifold-based variational optimization over centrally odd scalar functions on SU(2)^N. Instead of explicit Hilbert space vectors, Hamilton-Zero uses manifold functions on which the Hamiltonian operator acts through Lie derivatives. These derivatives are evaluated with specialized automatic differentiation primitives.

Density Matrix Representation Instead of Wave Function

Hamilton-Zero does not represent a single wave function, but an incoherent convex mixture of routed states as a density matrix ρ_θ. A Hamiltonian-conditioned routing policy π_θ returns probabilities, not amplitudes. With each evaluation, a new route is drawn, enabling adaptive structural adaptation to different Hamiltonian characteristics.

The Peter-Weyl theorem guarantees that the resulting variational principle preserves the ground state upper bound of the spin-1/2 sector. The architecture contains per-site tensors (q_t, h_t, ℓ_t) and per-bond tensors (J_tu, e_tu) that are relabeled route-dependently. A masking mechanism tracks real versus virtual positions.

Training Combines Techniques from LLMs and Reinforcement Learning

Training combines methods from Large Language Models with Deep Reinforcement Learning. The researchers developed an SU(2) Replica-Exchange Langevin Sampler and extended the Kronecker-Factored Approximate Curvature (KFAC) optimizer for this context. The model uses Sharded Natural-Gradient Optimization.

The training dataset varies systematically over connection topology, system size, interaction types and strengths. The researchers integrated literature from a century of many-body physics. Fine-tuning on a generalization dataset was performed for systems up to 1024 qubits.

Neural Tensor Network Readout Structure

The architecture uses a hierarchical structure with multiple levels:

• Leaf Builder and Contextualizer process per-site information
• A Merge Tree combines local information hierarchically
• A Root Readout generates final predictions

This approach enables simulation of disordered systems, estimation of the quantum geometric tensor in coupling space, and unsupervised detection of quantum phase transitions. The amortized computation across different Hamiltonian classes addresses a classically non-simulable problem by distributing learning costs across many systems.

Development Trend Toward Foundation Neural-Network Quantum States

The model follows a trend toward Foundation Neural-Network Quantum States (FNQS). In contrast to traditional Neural Quantum States, this approach processes multimodal inputs – both physical configurations and Hamiltonian couplings. Related work shows that trained neural networks can generate ground states for unseen Hamiltonian parameters without additional training.

The publication appeared on August 12, 2026 on arXiv (2608.11911) by Timothy Heightman and two other authors. A second version was published on August 13, 2026.