What does "${\cal M}_{A,\alpha}$ is a measurement operation" mean?

Question

My question regards this paper: https://arxiv.org/abs/1909.07534

If you look at the sentence below equation 8, it says that

Ma,x is measurement operation post-selected with a measurement outcome $x$.

What does it mean to post select? How is it implemented?

More specifically, figure 2 shows the decomposition of the CZ gate. The $(I + xZ)/2$ is a referred to as

Measurement operation that is postselected with measurment outcome x.

How can I implement this in Qiskit?

Please note that I am using $x$ instead of $\alpha$.

Answer 1

Answering only the first of these two questions: What does it mean to post select? How is it implemented?

Post selection relates to the fact that, after making the measurement you may be able to select between continuing the procedure and study the state, or discard the resulting system and start again the process.

It is implemented using classical feedforward in general, as once you have obtained an outcome from the measurement, you may decide if continuing with the protocol or discard the unwanted result and start again.

For instance, as shown in figure 2, this might be used to avoid 2-qubit gate implementation errors at the cost of probabilistic decomposing the gate into a single-qubit instances (and putting the overhead in the number of discarding operations needed, probability of post-selection, etc.).