[krastanov]

The magnitude of an "amplitude" is basis dependent. A basis is a human invention, an arbitrary choice made by the human to describe nature. The choice of basis is not fundamental. So just choose a basis in which there are no vanishingly small amplitudes and your worry is addressed.

[tromp]

Any implementation of Shor will need vanishingly small amplitudes, as it forms a superposition of 2^256 classical states.

[krastanov]

This is completely missing the point. There is nothing fundamental to an amplitude. The amplitudes are this small because you have chosen to work in a basis in which they are small. Go to the Hadamard basis and the amplitude value is exactly 1. After all, the initial state of Shor's algorithm (the superposition of all classical bitstrings) is the perfectly factorizable, completely not entangled state |+++++++>

[oh_my_goodness]

1/sqrt(N)