Two-dimensional local Hamiltonian problem with area laws is QMA-complete

TL;DR

This paper proves that determining the ground state energy of 2D local Hamiltonians with area laws is QMA-complete, indicating high computational complexity even under area law constraints.

Contribution

It establishes the QMA-completeness of the 2D local Hamiltonian problem with area laws, extending complexity results to translation-invariant and certain 3D models.

Findings

2D local Hamiltonian problem with area laws is QMA-complete

Ground states of 2D local Hamiltonians with area laws lack efficient classical representations

Complexity remains high even if area laws are proved for gapped systems

Abstract

We show that the two-dimensional (2D) local Hamiltonian problem with the constraint that the ground state obeys area laws is QMA-complete. We also prove similar results in 2D translation-invariant systems and for the 3D Heisenberg and Hubbard models with local magnetic fields. Consequently, unless MA = QMA, not all ground states of 2D local Hamiltonians with area laws have efficient classical representations that support efficient computation of local expectation values. In the future, even if area laws are proved for ground states of 2D gapped systems, the computational complexity of these systems remains unclear.