  We investigate the quantum parameter estimation in circuit quantum
electrodynamics via dispersive measurement. Based on the Metropolis Hastings
(MH) algorithm and the Markov chain Monte Carlo (MCMC) integration, a new
algorithm is proposed to calculate the Fisher information by the stochastic
master equation for unknown parameter estimation. Here, the Fisher information
is expressed in the form of log-likehood functions and further approximated by
the MCMC integration. Numerical results demonstrate that the single evolution
of the Fisher information can probably approach the quantum Fisher information.
The same phenomenon is observed in the ensemble evolution in the short time
interval. These results demonstrate the effectiveness of the proposed
algorithm.
