  In this paper, we construct a quadrature scheme to numerically solve the
nonlocal diffusion equation $(\mathcal{A}^\alpha+b\mathcal{I})u=f$ with
$\mathcal{A}^\alpha$ the $\alpha$-th power of the regularly accretive operator
$\mathcal{A}$. Rigorous error analysis is carried out and sharp error bounds
(up to some negligible constants) are obtained. The error estimates include a
wide range of cases in which the regularity index and spectral angle of
$\mathcal{A}$, the smoothness of $f$, the size of $b$ and $\alpha$ are all
involved. The quadrature scheme is exponentially convergent with respect to the
step size and is root-exponentially convergent with respect to the number of
solves. Some numerical tests are presented in the last section to verify the
sharpness of our estimates. Furthermore, both the scheme and the error bounds
can be utilized directly to solve and analyze time-dependent problems.
