  Kerr parametric oscillators (KPOs) have attracted increasing attention in
terms of their application to quantum information processing and quantum
simulations. The state preparation and measurement of KPOs are typical
requirements when they are used as qubits. The methods previously proposed for
state preparations of KPOs utilize modulation of a pump field or an auxiliary
drive field. We study the stochastic state preparation of a KPO based on
homodyne detection, which does not require modulation of a pump field nor an
auxiliary drive field, and thus can exclude unwanted effects of possible
imperfection in control of these fields. We quantitatively show that the
detection data, if averaged over a proper time to decrease the effect of
measurement noise, has a strong correlation with the state of the KPO, and
therefore can be used to estimate the state of the KPO (stochastic state
preparation). We examine the success probability of the state estimation taking
into account the effect of the measurement noise and bit flips. Moreover, the
proper range of the averaging time to realize a high success probability is
obtained by developing a binomial-coherent-state model, which describes the
stochastic dynamics of the KPO under homodyne detection.
