  The local Hamiltonian problem consists of estimating the ground-state energy
(given by the minimum eigenvalue) of a local quantum Hamiltonian.
  First, we show the existence of a good product-state approximation for the
ground-state energy of 2-local Hamiltonians with one or more of the following
properties: (1) high degree, (2) small expansion, or (3) a ground state with
sublinear entanglement with respect to some partition into small pieces. The
approximation based on degree is a surprising difference between quantum
Hamiltonians and classical CSPs (constraint satisfaction problems), since in
the classical setting, higher degree is usually associated with harder CSPs.
The approximation based on low entanglement, in turn, was previously known only
in the regime where the entanglement was close to zero. Since the existence of
a low-energy product state can be checked in NP, the result implies that any
Hamiltonian used for a quantum PCP theorem should have: (1) constant degree,
(2) constant expansion, (3) a "volume law" for entanglement with respect to any
partition into small parts.
  Second, we show that in several cases, good product-state approximations not
only exist, but can be found in polynomial time: (1) 2-local Hamiltonians on
any planar graph, solving an open problem of Bansal, Bravyi, and Terhal, (2)
dense k-local Hamiltonians for any constant k, solving an open problem of
Gharibian and Kempe, and (3) 2-local Hamiltonians on graphs with low threshold
rank, via a quantum generalization of a recent result of Barak, Raghavendra and
Steurer.
  Our work introduces two new tools which may be of independent interest.
First, we prove a new quantum version of the de Finetti theorem which does not
require the usual assumption of symmetry. Second, we describe a way to analyze
the application of the Lasserre/Parrilo SDP hierarchy to local quantum
Hamiltonians.
