  We perform highly accurate density matrix renormalization group (DMRG)
simulations to investigate the ground state properties of the spin-1/2
antiferromagnetic square lattice Heisenberg $J_1$-$J_2$ model. Based on studies
of numerous long cylinders with circumferences of up to 14 lattice spacings, we
obtain strong evidence for a topological quantum spin liquid state in the
region $0.41\leq J_2/J_1\leq 0.62$, separating conventional N\'eel and striped
antiferromagnetic states for smaller and larger $J_2/J_1$, respectively. The
quantum spin liquid is characterized numerically by the absence of magnetic or
valence bond solid order, and non-zero singlet and triplet energy gaps.
Furthermore, we positively identify its topological nature by measuring a
non-zero topological entanglement entropy $\gamma=0.70\pm 0.02$, extremely
close to $\gamma=\ln(2) \approx 0.69$ (expected for a $Z_2$ quantum spin
liquid) and a non-trivial finite size dimerization effect depending upon the
parity of the circumference of the cylinder. We also point out that a valence
bond solid, and indeed any discrete symmetry breaking state, would be expected
to show a constant correction to the entanglement entropy of {\sl opposite}
sign to the topological entanglement entropy.
