Maybe a different way to put it is that physical unitaries are never completely blackbox.

]]>Anyway, those great examples, the quantum switch and controlled unitaries, convincingly show that the standard circuit model should be extended.

]]>But if you consider the physical unitary transformation generated by evolving the (effective) Hamiltonian for a given amount of time, this unitary will have a well-defined phase, so it might be tempting to take this as the “true” phase of the unitary. There are two problems with it: the first is that we know that a constant shift in the energy of the Hamiltonian has no effect on the dynamics, precisely because it just changes the global phase of the unitary. And since the Hamiltonian is just effective anyway, there’s no reason to take seriously which constant shift is applied to it. The second problem is that if we do make this unitary controlled, in order to be able to measure the phase, we see that the phase we’ll see depends on how precisely we do the controlling. In the interferometric case, it will not depend only on the waveplates we are using, but also on the length of the arms of the interferometer.

But lets say you find a canonical energy shift to apply to the Hamiltonian, and mod out the phase you get from the specific implementation of the controlled unitary, so that you do have a definition of the global phase of the unitary. Is this enough to find out the global phase of a quantum state? Actually not! Because even if you know all the phases that were applied to it in the circuit, still there’s nothing telling you what was the state’s initial global phase.

Also, remember that charge conservation comes from Noether’s theorem precisely by saying that the global phase of a quantum state makes no difference for its dynamics, so it’s not as if we would ever be able to find it out. I think the best way to describe this ghostly phase is really to say that it is unphysical. You can also describe quantum states via density matrices and unitaries by their Choi operators, where this phase never shows up.

]]>Also this raises a question about the notion of a quantum state, which doesn’t depend on a global phase. If we can track “phases” of every quantum gate in a circuit then we should know the exact “phase” of the output. So, maybe quantum states physically are indeed vectors and not rays, we just can’t feel the difference in a common scenarios.

In other words, global phase factors are not unphysical, they are just hidden and inaccessible.

Sorry if this a little bit off-topic.

]]>As for your question, indeed if you have $U$ just as a non-controlled unitary in your circuit, then its phase will not have any physical effect, so we say that for any phase $\varphi$ the matrix $1\oplus e^{i\varphi}U$ is a valid controlled version of $U$. Now when you have a physical realisation of the controlled unitary the phase will have a physical effect, and will be fixed by the details of the experiment. For example in the interferometric realisation it will be determined by the relative phase between the arms of the interferometer. As far as I’m aware this phase you never actually matter, though, for example in Shor’s algorithm you do phase estimation of the modular exponentiation unitary, but only phase differences will be relevant for the result.

]]>Though I’m puzzled about one thing. If we have an unknown unitary then multiplying it by a phase doesn’t change its action. But controlled version of a unitary does depend on such phase. In other words, U and -U are indistinguishable blackboxes, but 1⊕U is not the same thing as 1⊕-U. How this is interpreted in the physical realization? ]]>

I would like to organize a conference, or at least a workshop in his honor. If anyone wants to collaborate, please send me an email.

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