Hamilton-Zero: A Neural Tensor-Network Foundation Model for Ground States of Arbitrary Quadratic Qubit Hamiltonians
Timothy Heightman, Elena Orlova, Philip Mantrov, Aleksei Ustimenko
quant-ph, cond-mat.dis-nn, cond-mat.str-el, cs.AI
2026-08-12
Trains a 547M-parameter foundation model on hundreds of thousands of quadratic qubit Hamiltonians; zero-shot inference compiles wavefunctions up to 8,100 qubits and fine-tuning brings median energy gaps to 1e-4 to 1e-2 percent of exact values.
Computing the ground state of a quantum many-body system is the workhorse calculation behind materials design, quantum chemistry, and combinatorial optimization. The ground state of N spins lives in a 2^N-dimensional complex space. Three classical routes each hit a wall: tensor networks excel in 1D but the area law turns against them in 2D, where required bond dimension grows with region boundaries; variational quantum algorithms suffer barren plateaus; and neural quantum states (NQS) reach 1,500-2,000 qubits but retrain a fresh network for every single system.
Earlier foundation-NQS attempts transferred across Hamiltonians but froze the interaction topology, varying only coupling coefficients. A fixed-length coupling vector cannot express a different interaction type or a different spin count. Cross-structure transfer requires the Hamiltonian to enter as a variable, typed interaction graph. Hamilton-Zero (from Simulacra Research, weights open-sourced) is the first to make that work across system size, topology, and interaction type simultaneously.
The core move changes the wavefunction's representation space. Instead of amplitudes over the discrete basis {up, down}^N, the model learns a function on the continuous manifold SU(2)^N to C. Each spin is a unit quaternion; the network consumes 4N continuous coordinates. Spin operators become left-invariant vector fields on this manifold, so the Hamiltonian turns into a second-order differential operator computable by automatic differentiation of the inputs. The interaction data (J, h) is a bare coupling tensor with no fixed length, which is what makes variable structure representable at all.
The manifold's function space is larger than the physical spin-1/2 Hilbert space, so an unconstrained network could leak into higher-spin sectors and report unphysical energies below the true ground state. The fix combines the Peter-Weyl theorem with two structural constraints: per-site central oddness and linearity in every quaternion. The function then provably stays in the spin-1/2 sector, the variational principle holds, and every reported energy is a rigorous upper bound on the true ground state.
Architecturally, two pathways meet at one controlled point. The Hamiltonian (J, h) flows through spectral normalization, a learnable featurizer, and L pre-norm Transformer blocks. The spin configuration enters only at the leaves, through an odd leaf builder that preserves linearity, followed by a balanced binary merge tree with one shared rank-4 contraction, a neural augmentation of tree tensor networks. Missing bonds use learned absent tokens rather than zeros, so "no connection" is itself a representable feature.
The cleverest layer is a deep-RL router. Merge-tree efficiency depends on which site goes to which leaf slot, which encodes the system's entanglement structure. A pointer-network policy decodes a site-to-slot permutation slot by slot, trained with score-function gradients using energy as reward. The model learns to pick a contraction tree per Hamiltonian.
The pretraining corpus has 5,000 interaction topologies spanning superconducting, trapped-ion, Rydberg, spin-qubit, and molecular platforms, expanded by two perturbation tiers: physics changes (add/noise/remove bonds, add fields) and exact gauge transformations (same physics, different coordinates), yielding hundreds of thousands of training systems hot-swapped each epoch.
Zero-shot (frozen weights, pure sampling) median energy gaps against exact diagonalization on three held-out sets: 4.02%, 4.21%, 12.63%. Fine-tuning trains only the compiled merge tree, about 4.7M parameters, under 1% of the full model, on a single A100:
| Setting | Zero-shot median gap | Fine-tuned median gap | Within 1% after fine-tune |
| Combinatorial opt (A) | 4.02% | 3.1e-4% | 90% |
| Unseen topologies (B) | 4.21% | 1.2e-3% | 94% |
| Hardest hold-out (C) | 12.63% | 6.0e-3% | 100% |
Large-system cases: 1,024-qubit weighted MaxCut recovers 96.2% of archived feasible cuts after fine-tuning; PPP conjugated carbon chains hold 96.1% of the infinite-chain scale at N=1,024; frustrated square lattices retain 84.9% at 2,025 spins, drop to 70.5% at 4,096, and fail at 8,100 (energy sign flips), which locates the current size-extrapolation boundary.
Cost side: one sample costs under 1e-6 USD, versus 4.25e-4 to 8.0e-2 USD per shot on Amazon Braket quantum hardware, a 425x to 80,000x gap. The released checkpoint cost roughly $30k to pretrain plus $30k for evaluation and fine-tuning. The pretraining scaling exponent is about C^-0.53, which the authors compare favorably to LLM scaling exponents.
Two bonus findings. Sweeping the Ising coupling parameter with frozen weights, fidelity susceptibility peaks near the critical point g=1, so the model locates a phase transition zero-shot. The router, on a 45x45 square lattice (2,025 spins, 30x the largest pretraining size), spontaneously organizes local 2x2 plaquettes into diagonal bands; at step 6, 75% of nearest-neighbor bonds share a 64-leaf cell versus 3.1% under random ordering, evidence of a transferable entanglement prior rather than memorized orderings.
For the quantum computing industry, this moves the baseline for useful quantum advantage. Classical ground-state solvers used to restart per system, so quantum hardware only had to beat cold-start optimization. Now the classical cost amortizes as 1/M across users of the checkpoint, and any advantage claim should compare total wall time and dollar cost against both zero-shot and equal-budget fine-tuned Hamilton-Zero on the same Hamiltonians and observables.
For AI practitioners, it is a full demonstration of the foundation-model playbook absorbing a new domain: LLM-style pretraining practice (perturbation against memorization, scaling laws), RL for discrete structural decisions, and second-order optimization (a KFAC extension) assembled on one physics problem. The engineering is itself an asset: the stock JAX stack would need over a year of wall time; their released kernels cut it to about four days.
The authors list three: padding to powers of two makes a 129-qubit system cost the same as 256; the order of automatic differentiation scales with Pauli-string weight, restricting the model to quadratic Hamiltonians (which is universal for quantum computation, but higher-order terms need ancilla compilation); and finite-sample Monte Carlo estimates can violate the variational bound through estimator noise or mixing bias, mitigated by burn-in and stationarity tests rather than guarantees.
Cold water from this reading: zero-shot clearly breaks above 4,096 spins, and the 8,100-qubit row has negative recovery, so "8,100 qubits" is a compilation demonstration more than usable accuracy. The MaxCut comparison uses archived feasible cuts, not certified optima; the PPP comparison uses thermodynamic-limit literature scales, not exact finite-size values, so the "96%" figures deserve discounting. Fine-tuning needs a single A100, but only on top of a $30k pretraining checkpoint that small teams can use yet not reproduce. The scaling exponent is measured in H200 hours rather than FLOPs and is hardware-dependent.