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Topology Obstructs Pure Foundation Neural Quantum States
Timothy Heightman, Elena Orlova, Philip Mantrov, Aleksei Ustimenko
TL;DR
Foundation neural quantum states aim to maintain positive overlap with the true ground state across an entire Hamiltonian family. The paper uses the topology of ground-state bundles to prove when continuous normalized vector models cannot sustain positive fidelity, and constructs an operator-valued alternative. For a topologically nontrivial ground-state line bundle, every continuous normalized vector model has a zero-fidelity parameter, where its energy is at least one spectral gap above the ground state.
Problem
Foundation neural quantum states aim to maintain positive overlap with the true ground state across an entire Hamiltonian family.
Source: p. 1Method
The paper uses the topology of ground-state bundles to prove when continuous normalized vector models cannot sustain positive fidelity, and constructs an operator-valued alternative.
Source: p. 2Results
For a topologically nontrivial ground-state line bundle, every continuous normalized vector model has a zero-fidelity parameter, where its energy is at least one spectral gap above the ground state.
Source: p. 2Takeaways & Limitations
The identity ˆH ˆP₀ = E₀ ˆP₀ yields an operator eigenvalue equation, with vectorization mapping ˆH ˆA₀ to (ˆI ⊗ ˆH)|ˆA₀⟩⟩.
Source: p. 4A nonzero top Chern class is sufficient but not necessary for the obstruction, and rank-r bundle nontriviality alone is insufficient when r > 1.
Source: p. 5
Abstract
from arXiv · showhide
Foundation models for ground states in spin-1/2 systems are a promising method for problems ranging from quantum chemistry to identifying new phase diagrams. Nearly all such models are currently pure-states that condition on the Hamiltonian's parameters, whose Monte Carlo samples give energy estimates according to the variational principle. In this contribution, we show that this representation is topologically obstructed. For any gapped Hamiltonian family whose ground-state bundle is non-trivial, every continuous normalized state-vector model has zero fidelity with the ground state at some parameter value in the Hamiltonian family. For that value, the energy is at least one spectral gap, $\Delta$, with an $O(\Delta)$ gap in an open-neighbourhood of that point. We show that this is a sufficient no-go also in the case of degenerate ground-state manifolds, time dynamics, and periodic systems with mixed space-time topology, demonstrating these obstructions on one- and two-qubit systems. We discuss how this causes a spike in the fidelity susceptibility, giving a numerical signature of a phase-transition where there is none. We then show that operator-valued models canonically avoid these obstructions and preserve topological information, implying a structural necessity in representation for foundation neural quantum states.
END MATTER
The end matter extends the pure-state obstruction to time evolution and periodic space–time, where continuous vector models must encounter zero-fidelity nodes. Matched operator-valued models avoid these nodes in the reported examples.
Numerical Comparisons
- The operator-valued model uses the same 2691-parameter architecture as the pure-state model and reaches a refined maximum infidelity of 1.5306 × 10^-6.Its reported energy error is 3.0613 × 10^-6.Source: END MATTER · p. 7
Dynamical Obstructions
- For every time slice, a continuous vector-valued model has zero fidelity at some Hamiltonian parameter when the initial ground-state bundle is nontrivial.The time-evolved line bundle remains isomorphic to the initial bundle at every time.Source: END MATTER · p. 7
Finite-Time Quench
- In the one-qubit quench, each time slice retains |c1| = 1, so the pure-state model’s zero becomes a charged worldline crossing every slice.The target forms a line bundle over S2 × [0, T].Source: END MATTER · p. 7
Periodic Systems
- A periodic family with c1(L) ≠ 0 forces every continuous periodic vector model to have a space–time fidelity node.No normalized true-state representative can be both global in x and periodic in t.Source: END MATTER · p. 7
Numerical Comparisons
- On the finite-time quench grid, the density-matrix-valued FNQS has maximum infidelity 2.000 × 10^-5, with no node forming in the compared slices.The paired models use identical architectures, optimizers, and training schedules; only the output map changes.Source: Figure 4 · p. 8
Periodic Systems
- The periodic one-qubit example shows a nonzero winding in k forcing a zero-fidelity node, while the operator-valued model is unobstructed.Continuity extends the failure over an open region of space–time.Source: END MATTER · p. 8
SUPPLEMENTAL MATERIAL
The supplemental results show that sampling does not remove the topological obstruction, while limited model capacity can produce additional neutral node pairs. The top-Chern-class criterion has scope limits, particularly for higher-rank bundles.
Limits of the Topological Criterion
- The top Chern class being nonzero is sufficient but not necessary for every section to vanish; for rank r > 1, bundle nontriviality alone is insufficient.Torsion or higher obstructions can also forbid a nowhere-zero section.Source: SUPPLEMENTAL MATERIAL · p. 9
Inexpressive Models Create Neutral Node Pairs
- A width-8 dynamical model produces up to two extra node pairs, while the total charge remains −1 on all 241 time slices.The node count passes 1 → 3 → 5 → 3 → 1; interior creation events add zero net charge.Source: SUPPLEMENTAL MATERIAL · p. 9, Figure S1 · p. 9
- Extra node pairs can be annihilated by added capacity and training, but the last node remains unless the output is operator-valued.The width-64 model carries one node per slice, whereas the width-8 model has additional pairs.Source: SUPPLEMENTAL MATERIAL · p. 9
Sampling Does Not Lift the Obstruction
- The obstruction constrains the model class rather than the estimator, so replacing exact energy and gradient evaluation with Monte Carlo sampling does not remove the node.The sampled wavefunction model retains the same forced node as the exact-training case.Source: SUPPLEMENTAL MATERIAL · p. 9, Figure S2 · p. 10
- Monte Carlo training preserves the wavefunction model’s forced zero-fidelity node, while a matched sampled density-matrix model reaches maximum infidelity 1.44 × 10^-6.Both models use N = 1024 samples per Hamiltonian per step.Source: SUPPLEMENTAL MATERIAL · p. 10, Figure S3 · p. 10