We study bifurcation behavior of a high-quality (high-Q) Josephson oscillator coupled to a superconducting qubit. It is shown that the probability of capture into the state of dynamic equilibrium is sensitive to qubit states. On this basis we present a new measurement method for the superposition state of a qubit due to its influence on transition probabilities between oscillator levels located in the energy region near the classical separatrix. The quantum-mechanical behavior of a bifurcation oscillator is also studied, which makes it possible to understand the mechanism of "entanglement" of oscillator and qubit states during the measurement process. The optimal parameters of the driving current and the state of the oscillator are found for performing one-qubit gates with the required precision, when the influence on the qubit from measurement back-action is minimal. A measurement protocol for state populations of the qubit entangled with the oscillator is presented.
Keywords: bifurcation; entanglement; measurement of superposition qubit states; nonlinear oscillator; qubit.