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I am interested in the execution time of quantum gates on IBM's $15$-qubits Melbourne computer. Is there any information regarding this?

After searching for a while I have only managed to find the times of the CNOT gates here, but unfortunately, no information for other quantum operations. I am specifically interested in the Hadamard gate .h() and the phase and rotation gates .rz() and .u3(), as well as the simple inverter .x(). I know these gates are not elementary (i.e they are composite gates) but still, any information regarding their respective execution times would be greatly appreciated!

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You can get all the information about execution times on a backend by using the .properties() method in Qiskit. In your example, this can be done as follows

backend = provider.get_backend('ibmq_16_melbourne')
backend.properties()

This returns an object containing all the gate times.

It is worth noting that all single qubit gates get converted into u3 gates, and all multiqubit gates converted to use cx, so that they can be run on the real hardware. If you want to see how a specific circuit is converted into this format you can use the transpiler, for example new_qc = transpile(qc, optimization_level=3)

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