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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

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  • $\begingroup$ What does it mean for a rewrite rule to be implementable by a circuit? For one thing, the bialgebra diagram of your post is equal to a linear map that is not unitary, so there is no unitary circuit that implements it. $\endgroup$
    – John
    Commented May 12, 2022 at 12:52
  • $\begingroup$ Hi @John, I know (now, after thinking a bit and seeing a presentation on gflow) that it is not a unitary, but I would like to understand if for instance we can implement using adaptivity for instance. Another aspect of the question, just as any other question, might be that it is just impossible to implement this, unitarily as it is clear or adaptively but without post selection. Either way, it might be just impossible to implement this map adaptively as well (maybe putting an ancilla, connecting with 'past' as in my last diagram and then adaptively resolve this in a future ancilla operation $\endgroup$
    – R.W
    Commented May 12, 2022 at 13:28
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    $\begingroup$ 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 $\endgroup$
    – John
    Commented May 12, 2022 at 13:49
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    $\begingroup$ If the direction of time is top to bottom, then the new diagram is a CNOT, which would be implementable without measurement. Maybe you have some additional constraints on how it has to be implemented, that prevent this? $\endgroup$ Commented May 14, 2022 at 18:12

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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.

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  • $\begingroup$ Just to to give some more intuition about why you can't implement non-isometries deterministically: each measurement outcome creates a branch that is a 'pure' postselected circuit, which hence sends pure states to pure states (proportionally). Since we are assuming determinism, all branches are equal and hence the overall circuit sends pure states to pure states. But this can then be described as a pure quantum process, and we know that everything pure in quantum mechanics is unitary. $\endgroup$
    – John
    Commented Sep 20, 2022 at 15:19

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