# The complexity of Area-lawed Hamiltonians

I have recently thought about "importing" some physics-related question into quantum CS:

The notion of the area-law phenomenon in Hamiltonian systems usually stands for a local Hamiltonian on some lattice, whose groundstate exhibits a property in which entanglement of any closed region is proportional to the surface of the region, and not its volume (as it would for a general state). A famous conjecture is whether all constant-gapped Hamiltonians exhibit this area-law property. For 1-dimensional systems, this question was answered in the positive by Hastings (arXiv:0705.2024).

Yet, the connection between such systems and complexity theory is very vague: while Hastings' result implies that 1-D area-law-abiding systems can be classically simulated, for general systems this is unknown. So my question is, is the quest to solve the area-law conjecture worthwhile? Or adversarially put, can one come up with a QMA-complete local Hamiltonian which is also area-law-abiding. A small glance at the known QMA-complete local Hamiltonians, which are essentially all based on Kitaev's quantum Cook-Levin theorem yields that these Hamiltonians do not have the area law property.

• Hit the Enter key too quickly? :) – Daniel Apon Nov 17 '10 at 12:06
• Please ignore my vote to close; Lior filled in the rest of his question! – Daniel Apon Nov 17 '10 at 12:27