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Why is the $|0\rangle$ state usually taken to be the ground state of a transmon qubit, and the $|1\rangle$ state the first excited state? Could you reverse this, and if so how would it affect the qubit's response to noise and 2-qubit gates?

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It would not make any difference whatsoever, as they are simply labels. "Red" and "blue" are equally valid names to give to qubits, albeit clunky ones.

A rose by any other name would smell just as sweet...

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