Quantum parameter estimation has many applications, from gravitational wave detection to quantum key distribution. We present the first experimental demonstration of the time-symmetric technique of quantum smoothing. We consider both adaptive and non
-adaptive quantum smoothing, and show that both are better than their well-known time-asymmetric counterparts (quantum filtering). For the problem of estimating a stochastically varying phase shift on a coherent beam, our theory predicts that adaptive quantum smoothing (the best scheme) gives an estimate with a mean-square error up to $2sqrt{2}$ times smaller than that from non-adaptive quantum filtering (the standard quantum limit). The experimentally measured improvement is $2.24 pm 0.14$.
