  We establish a quantitative version of the Tracy--Widom law for the largest
eigenvalue of high dimensional sample covariance matrices. To be precise, we
show that the fluctuations of the largest eigenvalue of a sample covariance
matrix $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$,
where $X$ is an $M \times N$ random matrix whose entries are independent real
or complex random variables, assuming that both $M$ and $N$ tend to infinity at
a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained
by Wang [73]. Our proof relies on a Green function comparison method [27] using
iterative cumulant expansions, the local laws for the Green function and
asymptotic properties of the correlation kernel of the white Wishart ensemble.
