  We implement, optimize, and validate the linear-scaling Kubo-Greenwood
quantum transport simulation on graphics processing units by examining resonant
scattering in graphene. We consider two practical representations of the
Kubo-Greenwood formula: a Green-Kubo formula based on the velocity
auto-correlation and an Einstein formula based on the mean square displacement.
The code is fully implemented on graphics processing units with a speedup
factor of up to 16 (using double-precision) relative to our CPU implementation.
We compare the kernel polynomial method and the Fourier transform method for
the approximation of the Dirac delta function and conclude that the former is
more efficient. In the ballistic regime, the Einstein formula can produce the
correct quantized conductance of one-dimensional graphene nanoribbons except
for an overshoot near the band edges. In the diffusive regime, the Green-Kubo
and the Einstein formalisms are demonstrated to be equivalent. A comparison of
the length-dependence of the conductance in the localization regime obtained by
the Einstein formula with that obtained by the non-equilibrium Green's function
method reveals the challenges in defining the length in the Kubo-Greenwood
formalism at the strongly localized regime.
