  We give a new proof for the area law for general 1D gapped systems, which
exponentially improves Hastings' famous result \cite{ref:Has07}. Specifically,
we show that for a chain of d-dimensional spins, governed by a 1D local
Hamiltonian with a spectral gap \eps>0, the entanglement entropy of the ground
state with respect to any cut in the chain is upper bounded by $O{\frac{\log^3
d}{\eps}}$. Our approach uses the framework Arad et al to construct a
Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable
factors. However, our construction uses the Hamiltonian directly, instead of
using the Detectability lemma, which allows us to work with general
(frustrated) Hamiltonians, as well as slightly improving the $1/\eps$
dependence of the bound in Arad et al. To achieve that, we establish a new,
"random-walk like", bound on the entanglement rank of an arbitrary power of a
1D Hamiltonian, which might be of independent interest: \ER{H^\ell} \le (\ell
d)^{O(\sqrt{\ell})}. Finally, treating d as a constant, our AGSP shows that the
ground state is well approximated by a matrix product state with a sublinear
bond dimension $B=e^{O(\log^{3/4}n/\eps^{1/4})}. Using this in conjunction with
known dynamical programing algorithms, yields an algorithm for a 1/\poly(n)
approximation of the ground energy with a subexponential running time T\le
\exp(e^{O(\log^{3/4}n/\eps^{1/4})}).
