  Estimating the left tail of quadratic forms in Gaussian random vectors is of
major practical importance in many applications. In this letter, we propose an
efficient importance sampling estimator that is endowed with the bounded
relative error property. This property significantly reduces the number of
simulation runs required by the proposed estimator compared to naive Monte
Carlo (MC), especially when the probability of interest is very small. Selected
simulation results are presented to illustrate the efficiency of our estimator
compared to naive MC as well as some of the well-known approximations.
