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Given that the global phases of states cannot be physically discerned, why is it that quantum circuits are phrased in terms of unitaries and not special unitaries? One answer I got was that it is just for convenience but I'm still unsure.

A related question is this: are there any differences in the physical implementation of a unitary $U$ (mathematical matrix) and $ V: =e^{i\alpha}U$, say in terms of some elementary gates? Suppose there isn't (which is my understanding). Then the physical implementation of $c\text{-}U$ and $c\text{-}V$ should be the same (just add controls to the elementary gates). But then I get into the contradiction that $c\text{-}U$ and $c\text{-}V$ of these two unitaries may not be equivalent up to phase (as mathematical matrices), so it seems plausible they correspond to different physical implementations.

What have I done wrong in my reasoning here, because it suggests now that $U$ and $V$ must be implemented differently even though they are equivalent up to phase?

Another related question (in fact the origin of my confusion, I'd be extra grateful for an answer to this one): it seems that one can use a quantum circuit to estimate both the modulus and phase of the complex overlap $\langle\psi|U|\psi\rangle$ (see https://arxiv.org/abs/quant-ph/0203016). But doesn't this imply again that $U$ and $e^{i\alpha}U$ are measurably different?

2It is more philosophically accurate to say projective unitary group $PU$ instead. That is because the operation is to take an arbitrary unitary matrix and lose the phase vs the subset for which that phase is $1$. The maps go $SU \to U \to PU$ so they are on opposite sides of the arrows. – AHusain – 2018-05-12T22:04:03.400

@AHusain Which are "The maps"? In terms of quotienting out, it will go $U\to SU\to PU$. – Norbert Schuch – 2018-05-13T10:48:53.400

1No. SU is the subset with determinant 1, so it includes with a map into U. PU is the quotienting out. You can take a projective unitary and give a representative in SU with determinant 1, but that is not automatic. – AHusain – 2018-05-14T15:31:27.743