  In this article we propose a method to estimate with high accuracy pure
quantum states of a single qudit. Our method is based on the minimization of
the squared error between the complex probability amplitudes of the unknown
state and its estimate. We show by means of numerical experiments that the
estimation accuracy of the present method, which is given by the expectation of
the squared error on the sample space of estimates, is state independent.
Furthermore, the estimation accuracy delivered by our method is close to twice
the Gill-Massar lower bound, which represents the best achievable accuracy, for
all inspected dimensions. The minimization problem is solved via the
concatenation of the Complex simultaneous perturbation approximation, an
iterative stochastic optimization method that works within the field of the
complex numbers, and Maximum likelihood estimation, a well-known statistical
inference method. This can be carried out with the help of a multi-arm
interferometric array. In the case of a single qubit, a Mach-Zehnder
interferometer suffices. We also show that our estimation procedure can be
easily extended to estimate unknown unitary transformations acting on a single
qudit. Thereby, the estimation of unitary transformations achieves a higher
accuracy than that achieved by processes based on tomographic methods for mixed
states.
