Graph-Theoretic Analysis of Phase Optimization Complexity in Variational Wave Functions for Heisenberg Antiferromagnets

2026-02-04

AI Summary

This work investigates the computational complexity of reconstructing the phase structure in variational wave functions for frustrated Heisenberg antiferromagnets. By modeling the Hilbert space as a weighted graph—termed the Hilbert graph—and restricting the phase degrees of freedom to ℤ₂ values while fixing the amplitudes, the ground-state phase reconstruction problem is mapped exactly onto a weighted Max-Cut problem on this graph. This approach establishes, for the first time, a direct connection between the sign structure of quantum many-body wave functions and combinatorial optimization. The authors prove that phase reconstruction is NP-hard in the worst case, thereby revealing the intrinsic computational difficulty underlying variational phase optimization and offering a new perspective on the sign problem in quantum many-body systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Social Networks and Social Media: Computational social science

Abstract

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.

Problem

• Heisenberg antiferromagnets
• phase structure
• variational wave functions
• frustrated magnetism
• Hilbert space

Innovation

• Hilbert graph
• phase optimization
• Max-Cut
• NP-hard
• variational wave functions