I'm curious, why L2 norm? Each qubit lives in C^2, does it not? I've only ever seen quantum computing formulated over finite dimensional Hilbert spaces (though to be fair it was only in a math context, no real physics there)
We can't have a why, because the claim that the norm is L2 is axiomatic [1]. However, as the sibling comment notes, we can argue from consistency. Would any other norm give us a sensible theory? Or we can put quantum theory in a larger framework of theories and discuss why the L2 norm makes more sense within that framework. Scott Aaronson discusses some of these idea in his lecture [2].
