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This paper establishes a rigorous theoretical link between variational quantum wavefunction optimization and combinatorial problems like Max-Cut, providing essential complexity bounds for training neural quantum states in condensed matter physics. By mapping Hilbert space structures to classical Ising models, the work clarifies the fundamental computational limits of learning ground state phases in Heisenberg antiferromagnets.
Graph-Theoretic Analysis of Phase Optimization Complexity in Variational Wave Functions for Heisenberg Antiferromagnets
arXiv:2602.04943v4 Announce Type: replace-cross Abstract: We study the computational complexity of learning the ground state phase structure of Heisenberg antiferromagnets. Representing Hilbert space as a weighted graph, the variational energy defines a weighted XY model that, for $\mathbb{Z}_2$ phases, reduces to a classical antiferromagnetic Ising model on that graph. For fixed amplitudes, reconstructing the signs of the ground state wavefunction thus reduces to a weighted Max-Cut instance. This links sign optimization of wavefunctions directly to combinatorial optimization.
Abstract:We study the computational complexity of learning the ground state phase structure of Heisenberg antiferromagnets. Representing Hilbert space as a weighted graph, the variational energy defines a weighted XY model that, for $\mathbb{Z}_2$ phases, reduces to a classical antiferromagnetic Ising model on that graph. For fixed amplitudes, reconstructing the signs of the ground state wavefunction thus reduces to a weighted Max-Cut instance. This links sign optimization of wavefunctions directly to combinatorial optimization.
Strongly Correlated Electrons (cond-mat.str-el); Disordered Systems and Neural Networks (cond-mat.dis-nn); Artificial Intelligence (cs.AI); Computational Complexity (cs.CC); Quantum Physics (quant-ph)
arXiv:2602.04943 [cond-mat.str-el]
(or arXiv:2602.04943v4 [cond-mat.str-el] for this version)
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