Controlled-NOT logic with nonresonant Josephson phase qubits

We establish theoretical bounds on qubit detuning for some of the previously proposed controlled-NOT (CNOT) logic gate implementations with weakly coupled Josephson phase qubits. It is found that in the two-step, root iSWAP-based case the value of the detuning during the entangling operations must not exceed 2g, where g is the characteristic coupling constant. In the single-step case we consider two practical, physically distinct implementations, in which one of the qubits is driven by a concurrent rf pulse of fixed frequency. We find that when the local drive is applied to the reference qubit (with which it is in resonance), the detuning should not exceed g. If the drive is applied to the detuned qubit, generation of the perfect CNOT gate is possible at any value of detuning provided that the amplitude of the pulse can be made arbitrarily large.