  Ground states of local Hamiltonians can be generally highly entangled: any
quantum circuit that generates them (even approximately) must be sufficiently
deep to allow coupling (entanglement) between any pair of qubits. Until now
this property was not known to be "robust" - the marginals of such states to a
subset of the qubits containing all but a small constant fraction of them may
be only locally entangled, and hence approximable by shallow quantum circuits.
In this work we construct a family of 16-local Hamiltonians for which any
1-10^{-9} fraction of qubits of any ground state must be highly entangled.
  This provides evidence that quantum entanglement is not very fragile, and
perhaps our intuition about its instability is an artifact of considering local
Hamiltonians which are not only local but spatially local. Formally, it
provides positive evidence for two wide-open conjectures in condensed-matter
physics and quantum complexity theory which are the qLDPC conjecture, positing
the existence of "good" quantum LDPC codes, and the NLTS conjecture due to
Freedman and Hastings positing the existence of local Hamiltonians in which any
low-energy state is highly-entangled.
  Our Hamiltonian is based on applying the hypergraph product by Tillich and
Zemor to a classical locally testable code. A key tool in our proof is a new
lower bound on the vertex expansion of the output of low-depth quantum
circuits, which may be of independent interest.
