  In contrast to the standard quantum state tomography, the direct tomography
seeks the direct access to the complex values of the wave function at
particular positions (i.e., the expansion coefficient in a fixed basis).
Originally put forward as a special case of weak measurement, it can be
extended to arbitrary measurement setup. We generalize the idea of "quantum
metrology," where a real-valued phase is estimated, to the estimation of
complex-valued phase, and apply it for the direct tomography of the wave
function. It turns out that the reformulation can help us easily find the
optimal measurements for efficient estimation. We further propose two different
measurement schemes that eventually approach the Heisenberg limit. In the first
scheme, the ensemble of measured system is duplicated and the replica ensemble
is time-reversal transformed before the start of the measurement. In the other
method, the pointers are prepared in special entangled states, either GHZ-like
maximally entangled state or the symmetric Dicke state. In both methods, the
real part of the parameter is estimated with a Ramsey-type interferometry while
the imaginary part is estimated by amplitude measurements.
