Hamilton-Zero: A Neural Tensor-Network Foundation Model for Ground States of Arbitrary Quadratic Qubit Hamiltonians

2026-08-12

AI Summary

This work proposes the first foundation model dedicated to quantum ground state computation, overcoming the scalability and accuracy limitations of classical approaches. For arbitrary quadratic qubit Hamiltonians, it formulates ground state learning as a variational optimization of central odd scalar functions on the SU(2)^N manifold, eschewing conventional Hilbert space representations in favor of Lie derivatives and custom automatic differentiation. The method integrates neural tensor networks, SU(2) replica-exchange Langevin sampling, block-wise natural gradients, and an extended KFAC optimizer within a pretraining–fine-tuning paradigm. Pretrained on hundreds of thousands of Hamiltonian systems, the model generalizes successfully to systems with up to 1,024 qubits and has been validated at scales reaching 8,100 qubits. Theoretical guarantees based on the Peter–Weyl theorem ensure the correctness of the variational upper bound, supporting arbitrary topologies, system sizes, and interaction types.

Abstract

A central promise of useful quantum advantage is the ability to compute ground states of Hamiltonian systems beyond the reach of classical simulation methods. Here we demonstrate that this problem can be effectively amortized across an arbitrary and universal set of Hamiltonians by a foundation model with $\sim0.5$B variational parameters, trained with contemporary techniques from large language models and deep reinforcement learning. To do this, we formulate $\text{spin-}1/2$ quantum ground-state learning as manifold variational optimisation over centrally odd scalar functions on $\mathrm{SU}(2)^N$. This replaces explicit Hilbert-space vector amplitudes with manifold functions on which the Hamiltonian acts through Lie derivatives, evaluated by custom automatic differentiation primitives. We prove that the resulting variational principle on this manifold preserves the $\text{spin-}1/2$ sector's ground-state upper bound using the Peter-Weyl theorem, then pre-train our foundation model on a dataset of hundreds of thousands of different Hamiltonian systems, varying the connection topology, system size, interaction types and strengths, bringing together a century of many-body literature. Using a novel $\mathrm{SU}(2)$ replica-exchange Langevin sampler and sharded natural-gradient optimisation, we train our model with our own extension of the Kronecker-Factored Approximate Curvature (KFAC) optimiser on system sizes up to 64 qubits. On a held-out generalisation dataset, we fine-tune our model on system sizes of up to 1024 qubits, and evaluate on systems up to 8100 qubits.

Problem

• ground states
• quadratic Hamiltonians
• quantum advantage
• qubit systems
• many-body physics

Innovation

• neural tensor network
• manifold variational optimization
• SU(2) Lie derivatives
• foundation model for quantum ground states
• Kronecker-Factored Approximate Curvature