We can measure the second qubit as early as we like, without changing the output from the circuit. In fact, we can do the measurement right at the beginning of the protocol, giving:pic.twitter.com/obUVj5snWf
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That's a measurement based uncomputation. If a qubit Q is a computational basis function of other qubits R, you can get rid of Q by measuring it in the X basis and using the result to control phasing operations on R. Useful tool. e.g. see Appendix C of https://arxiv.org/abs/1902.02134
That appendix doesn't seem to address my question. Note that I'm extremely familiar with the facts you mention - I've written papers based on those observations (and a review: https://arxiv.org/pdf/quant-ph/0504097.pdf … ). But none of that gives a conceptual explanation.
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The X basis measurement works because the last thing that happened to Q was being targeted by a CNOT. This is an X basis interaction. So an X basis measurement commutes with this operation. Allows you to use the deferred measurement principle, leaving no 2-qubit ops on Q.
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X does not commute with the CNOT control, which this is.
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