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I was wondering why IBM's computer were using U1 and U2 gates as part of there basis gates since as I understood, they are particular cases of the U3 gate. Why not just use U3 gate instead ?

Thanks in advance

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There is not anything that you cant do with U3 so ideally there is no reason for U1 and U2. Eventually, as the transpilers gets better we may remove them and just have U3 and CNOT. So why did we make U1 and U2? It is because of the hardware. The U1 is done using a frame change (see https://arxiv.org/abs/1612.00858) which means they are done in software (gate time is zero). U2 is a single $\pi/2$ gate with a pre and post frame change (gate time is one unit of single-qubit gates). The one we calibrate is a $\pi/2$ rotation around the x-axis and choose the pre and post to get the definition of U2. The U3 is three frame changes and two-single qubit gates.

The reason why you would use them is the error rate. For the U1 the erorr rate is not measurable (so let us say it is zero). The error for the U3 is twice that of the U2. So if you want a better fidelity (ignoring spectator qubit errors - these are errors from parallel operations or spectator qubits being in a state that is not zero) you want to minimize the number of gates with errors so a circuit of just U1 will be better than one using U3.

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