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Optimal state estimation for linear discrete-time systems is considered.Motivated by the literature on differential privacy, the measurements areassumed to be corrupted by Laplace noise. The optimal least mean square errorestimate of the state is approximated using a randomized method. The methodrelies on that the Laplace noise can be rewritten as Gaussian noise scaled byRayleigh random variable. The probability of the event that the distancebetween the approximation and the best estimate is smaller than a constant isdetermined as function of the number of parallel Kalman filters that is used inthe randomized method. This estimator is then compared with the optimal linearestimator, the maximum a posteriori (MAP) estimate of the state, and theparticle filter.

Optimal State Estimation with Measurements Corrupted by Laplace Noise