In a flux qubit, the state of a qubit can be changed using flux. Therefore, to provide a data to the qubit we have to apply external flux. It is possible by a mutual inductor. SQUID in flux qubit induced by a external inductor. Actually,  mutual induction here used as classical and quantum interface. Even it is used to tune the qubit with cavity resonator.

A flux qubit, which is a superconducting loop interrupted by ultrasmall Josephson junctions, can be coupled inductively by using the flux generated circulating currents in the loop. Experiments on coupled flux qubits employed fixed coupling through mutual inductance. It is very desirable for such coupling to be switchable with a fast switching time in order to realize efficient operation for multiple qubits.

In addition to previous answers, I can add that the mutual inductance is very important in many superconducting qubit designs, when the coupling to the qubit is inductive. It sets the coupling strength between adjacent qubits or between the qubit and a quantum bus such as a resonator. In that ladder case it thus govern the observers ability to interact with the qubit.