Topology limits pure state models in quantum ground state prediction
Topology Obstructs Pure Foundation Neural Quantum States
Artificial Intelligence
Summary
Predicting the lowest energy states of quantum systems is important for fields like chemistry and materials science. This paper shows that for some quantum systems with complicated underlying structures, models that use pure quantum states to represent these lowest energy states cannot always match the true states perfectly. The authors reveal that this is due to topological features that create unavoidable mismatches at certain parameter values, causing errors in energy estimates. They suggest that using more complex mathematical objects called operator-valued models can overcome these issues and better capture the necessary information.
quantum ground statespin-1/2 systemstopologypure statevariational principleHamiltonianspectral gapfidelityphase transitionoperator-valued models
Authors
Timothy Heightman, Elena Orlova, Philip Mantrov, Aleksei Ustimenko
Abstract
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, $Δ$, with an $O(Δ)$ 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.