[oh_my_goodness]

I don’t think we can discuss precision usefully without numbers. We seem to agree on the word “finite” but that covers a lot of ground. “High” precision in rotating an optical polarization to exactly 45 degrees is maybe 60 dB, +- 0.0001% of the probability. That means the amplitudes are matched within 0.1%. 0.1% is fine for two qubits with 4 states. It might work for 8 qubits (256 states). For 256 qubits, no.

[oh_my_goodness]

Gah, I wrote the wrong thing. If each probability is 50% +- 10^{-6} then the amplitudes are matched to within around 2 times 10^{-6}.

But when N>2 this gets tougher rapidly.

If we add 10^12 complex amplitudes and each one is off by one part in 10^{-6}, we could easily have serious problems with the accuracy of the sum. And 10^12 amplitudes is "only" around 40 qubits.