[MontyCarloHall]

I doubt it. Shor's algorithm relies on the quantum Fourier transform, which requires the complex phase information encoded in the quantum wavefunctions. The quantum probability norm (L2) accounts for interference between the complex amplitudes of these wavefunctions; the classical L1 probability norm does not.

[ogogmad]

I'm not sure that it's just L1 vs L2, since the Wigner formulation of quantum mechanics uses real-valued quasi-probabilities, but ones which can take negative values.

Oh, and also, if you swap out h-bar in Wigner's equations with some wavelength \lambda, you can interpret it in terms of classical wave optics... somehow. I'm not sure.