  A novel unitary quantum lattice gas algorithm is used to simulate quantum
turbulence of a BEC described by the Gross-Pitaevskii equation on grids up to
5760^3. For the first time, an accurate power law scaling for the quantum
Kelvin wave cascade is determined: k^{-3}. The incompressible kinetic energy
spectrum exhibits very distinct power law spectra in 3 ranges of k-space: a
classical Kolmogorov k^{-5/3} spectrum at scales much greater than the
individual quantum vortex cores, and a quantum Kelvin wave cascade spectrum
k^{-3} on scales of order the vortex cores. In the semiclassical regime between
these two spectra there is a pronounced steeper spectral decay, with
non-universal exponent. The Kelvin k^{-3} spectrum is very robust, even on
small grids, while the Kolmogorov k^{-5/3} spectrum becomes more and more
apparent as the grids increase from 2048^3 grids to 5760^3.
