  The accuracy of estimating $d$-dimensional quantum states is limited by the
Gill-Massar bound. It can be saturated in the qubit ($d=2$) scenario using
adaptive standard quantum tomography. In higher dimensions, however, this is
not the case and the accuracy achievable with adaptive quantum tomography
quickly deteriorates with increasing $d$. Moreover, it is not known whether or
not the Gill-Massar bound can be reached for an arbitrary $d$. To overcome this
limitation, we introduce an adaptive tomographic method that is characterized
by a precision that is better than half that of the Gill-Massar bound for any
finite dimension. This provides a new achievable accuracy limit for quantum
state estimation. We demonstrate the high-accuracy of our method by estimating
the state of 10-dimensional quantum systems. With the advent of new
technologies capable of high-dimensional quantum information processing, our
results become critically relevant as state reconstruction is an essential tool
for certifying the proper operation of quantum devices.
