Complexity Classification of Product State Problems for Local Hamiltonians
John Kallaugher, Ojas Parekh, Kevin Thompson, Yipu Wang, Justin Yirka
TL;DR
This work delivers a complete complexity dichotomy for optimizing minimum energy over product states for fixed 2-local Hamiltonian terms: the product-state problem lies in $\\mathsf{P}$ if all interactions are 1-local, and is $\\mathsf{NP}$-complete otherwise, with hardness persisting under constant-magnitude couplings. The authors introduce the $W$-linear Max-Cut framework, proving $MC^{L}_{W}$ is NP-complete for fixed non-negative diagonal $W$, and use gadget-based reductions to embed these vector-cut objectives into product-state energies. They then apply these reductions to show NP-hardness for product-state versions of Quantum Max-Cut and for $MC_{3}$, even in unweighted/positive-weight settings, thereby linking product-state optimization tightly to classic and quantum Hamiltonian complexity. The results imply that product-state approaches can inherit the hardness of full ground-state problems, while also enriching the optimization literature with new vector-embedding hard problems. Overall, the paper advances our understanding of when product-state ansätze yield efficiently solvable problems and provides new tools (gadget reductions and stretched vector cuts) of independent interest for constraint-satisfaction and optimization theory.
Abstract
Product states, unentangled tensor products of single qubits, are a ubiquitous ansatz in quantum computation, including for state-of-the-art Hamiltonian approximation algorithms. A natural question is whether we should expect to efficiently solve product state problems on any interesting families of Hamiltonians. We completely classify the complexity of finding minimum-energy product states for Hamiltonians defined by any fixed set of allowed 2-qubit interactions. Our results follow a line of work classifying the complexity of solving Hamiltonian problems and classical constraint satisfaction problems based on the allowed constraints. We prove that estimating the minimum energy of a product state is in P if and only if all allowed interactions are 1-local, and NP-complete otherwise. Equivalently, any family of non-trivial two-body interactions generates Hamiltonians with NP-complete product-state problems. Our hardness constructions only require coupling strengths of constant magnitude. A crucial component of our proofs is a collection of hardness results for a new variant of the Vector Max-Cut problem, which should be of independent interest. Our definition involves sums of distances rather than squared distances and allows linear stretches. We similarly give a proof that the original Vector Max-Cut problem is NP-complete in 3 dimensions. This implies that optimizing over product states for Quantum Max-Cut (the quantum Heisenberg model) is NP-complete, even when every term is guaranteed to have positive unit weight.