Implementation of two qubit gate decomposition in local operations

Question

My questions is regards to this paper: https://arxiv.org/abs/1909.07534

The above is the decomposition of a two qubit gate into local operations. Please note they are using a Super Operator formalism.

I am trying to implement the decomposition in qiskit. In appendix B, they list out the 6 operations that are needed, but say that the operations are not physical when $\beta=-1$ but achievable via classical post processing. See image below:

For the operations $\phi_{3,b}$ to $\phi_{6,b}$ they have $\beta M_{A_1,\beta}$. How is this operations actually implemented? My understanding is that I have a measurement operations in the circuit that after running multiple shots records that probability of getting either a +1 or -1. I then calculate 2 expectation values from this circuit for when the state was projected to $|0>$ and $|1>$.These expectation values have to then be multiplied by the probability and then summed together along with the rest of the terms in the equation to match the expectation of the non-decomposed circuit.

Is my reasoning correct?

Best regards and thanks in advance.

Answer 1

$\mathcal{M}_{A,\beta}$ is just a projective measurement onto the basis $A$. We would normally think about implementing this by defining $U$ to be the unitary that diagonalises $A$: $UAU^\dagger=D$. In that case, the measurement is performed by applying the operation $U$, performing a measurement in the standard basis and, if you get the desired answer $\beta$, applying $U^\dagger$. If you don't get the desired answer, you just throw the system away (unless you know how to convert from the incorrect measurement result to the correct one, which you might, depending on what you're doing).