Graph--Theoretic Analysis of Phase Optimization Complexity in Variational Wave Functions for Heisenberg Antiferromagnets
This work addresses a fundamental challenge in quantum magnetism for physicists, providing a direct link between wavefunction sign structure and combinatorial optimization, but it is incremental as it builds on existing variational methods.
The authors tackled the problem of characterizing the phase structure of wavefunctions in frustrated Heisenberg antiferromagnets by representing the Hilbert space as a weighted graph, showing that phase reconstruction reduces to a weighted Max-Cut instance, which is worst-case NP-hard.
Despite extensive study, the phase structure of the wavefunctions in frustrated Heisenberg antiferromagnets (HAF) is not yet systematically characterized. In this work, we represent the Hilbert space of an HAF as a weighted graph, which we term the Hilbert graph (HG), whose vertices are spin configurations and whose edges are generated by off-diagonal spin-flip terms of the Heisenberg Hamiltonian, with weights set by products of wavefunction amplitudes. Holding the amplitudes fixed and restricting phases to $\mathbb{Z}_2$ values, the phase-dependent variational energy can be recast as a classical Ising antiferromagnet on the HG, so that phase reconstruction of the ground state reduces to a weighted Max-Cut instance. This shows that phase reconstruction HAF is worst-case NP-hard and provides a direct link between wavefunction sign structure and combinatorial optimization.