How does the phase kickback work in real hardware?

Question

I have a trouble understanding the phase kickback, this figure from Kaye-Laflamme-Mosca's book summarizes the idea concisely;

Here is what I find:

When you do the calculations, you find that the phase is attached to the bottom qubit, somehow you are allowed to take this factor up to the top qubit, while this is possible by the axioms of the tensor product ( $|\alpha u\rangle|v\rangle =|u\rangle|\alpha v\rangle$ ) I am wondering how this is implemented in hardware?

The 'wires' represents qubits, how does a physical qubit 'transfer' its phase to another qubit? Or how do we 'force' this?

Answer 1

It probably helps to think about the controlled phase gate $$ CP=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}. $$ You should really interpret this as "change the phase of the state if it's in $|11\rangle$". It's a collective operation, not localised to a single qubit.

Yes, it is convenient for us to think about this gate as controlled-$Z$: apply $Z$ to the target if the control is 1, and that leads us to thinking like the phase is being applied just to the target qubit. But that's the convenient classical description, not what it's really doing. It is a global operation that makes no distinction between which of the two subsystems the phase is applied to. This is perhaps emphasised even more strongly in this case in the sense that controlled-phase is symmetric - it doesn't matter which qubit you call the control, and which the target!

Answer 2

If, as is here, the target qubit is prepared in $|−\rangle=\frac{|0\rangle-|1\rangle}{\sqrt 2}$, then the phase is kicked back to the control qubit merely by operation of the controlled-$\hat{U}$ gate. The stuff that's in the dashed-circles of the figure just "happens". There's no special operation, apart from the controlled-$\hat U$ gate, that's needed.

But if your question is how the controlled-$\hat{U}$ gate is actually implemented, then this is hardware-specific. For example trapped ions (I think) use sophisticated versions of a Cirac-Zoller gate, while neutral atoms use a Rydberg blockade, and superconducting qubits use microwave-pulses. Given some (single-qubit) gate $\hat U$, it's mostly a purely mechanical procedure to convert this gate into a sequence of laser pulses or microwave pulses to make it into a controlled-$\hat U$.