This protocol is extracted from research article:
Qubit parity measurement by parametric driving in circuit QED
Sci Adv, Nov 30, 2018; DOI: 10.1126/sciadv.aau1695

To model the back action of the homodyne measurement chain, we simulated multiple realizations of the evolution of the system under the stochastic master equation (24)Embedded Image(9)where Embedded Image is the dissipation superoperator and Embedded Image is the homodyne measurement back-action superoperator. Moreover, dW is a Wiener increment, which has statistical properties E[dW] = 0, E[dW2] = dt, with E[•] denoting the ensemble average. The results of Fig. 2 were obtained using Eq. 9 with the Hamiltonian Eq. 5. Equation 9 shows that the Hamiltonian and dissipation (first two terms) are symmetric under the transformation â → − â. This symmetry is broken by the homodyne measurement back action (last term), that is, by conditioning the state on the measurement record. In other words, although the average displacement of the resonator is null, conditioning the state on the measurement record makes it collapse onto ± αo.

The homodyne current resulting from the stochastic master equation is given by Embedded Image. For a given measurement time τ, the dimensionless integrated signal is given by Embedded Image.

To focus solely on the measurement scheme itself, we considered a homodyne measurement chain with unit efficiency. Because of the large number of photons in steady state, |αo|2, the measurement time and fidelity are mostly limited by the bifurcation time. As a result, adding imperfections to the measurement line affects the measurement time in a negligible manner. Moreover, for the parameters considered in the main text, the output power to amplify is below the 1-dB compression point for state-of-the-art amplifiers (42).

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