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

TL;DR

This paper demonstrates that reconstructing the ground state phase structure of Heisenberg antiferromagnets is computationally NP-hard by linking it to classical combinatorial optimization problems.

Contribution

It establishes the NP-hardness of phase reconstruction in Heisenberg antiferromagnets via a graph-theoretic approach, connecting quantum physics to classical optimization.

Findings

Reconstruction reduces to a weighted Max-Cut problem.

Ground state phase reconstruction is NP-hard in the worst case.

The analysis links quantum state learning to classical combinatorial complexity.

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 establishes that ground state phase reconstruction for Heisenberg antiferromagnets is worst-case NP-hard and links the task to combinatorial optimization.