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In the ZX-calculus, one of the fundamental rules of the diagrammatic reasoning is known as the bialgebra rule and it is described by the given diagrammatic equation:

enter image description here

Question: Can we implement this diagram without using post-selection or adaptivity?

I know that this rule preserves the gflow of a diagram, which implies that if a circuit is transformed to a ZX-calculus diagram and after simplification this rule is used then the resulting ZX-diagram can be transformed back into a quantum circuit. But from what I could understand, the bialgebra seems to preserve the gflow but it is not directly circuit implementable (i.e. with unitaries) because there is a $2 \to 1 \to 2$ qubit flow that must result in measurement. Hence, any non-adaptive and non-postselected implementation likely must come from not using single two systems but a possibly more complex system.

The adaptive implementation of the bialgebra (left side of the diagram) is very interesting but I think that it would be more interesting to have a circuit implementing. Moreover there is the $a\oplus_2b\pi $ term in the beggining that is very complicated to get rid of.

In this question, notation and reference to known results were taken from Ref. ZX-calculus for the working quantum computer scientist. The last diagram is mine so it might have mistakes.

enter image description here

R.W
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1 Answers1

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It is not possible to implement this operation without post-selection nor adaptivity. As I had described in my attempt, this kind of diagram does not represent a unitary, and although with post-selection it is possible the answer given by @John in the comments completely settles the question.

His remark might be relevant for future users though:

It is not possible to implement anything that is not an isometry in a deterministic way, even allowing for adaptation. You need to allow for different branches to have different outcomes.

R.W
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