  We study the basic features of the two-dimensional quantum Hubbard Model at
half-filling by means of the L\"uscher algorithm and the algorithm based on
direct update of the determinant of the fermionic matrix. We implement the
L\"uscher idea employing the transfer matrix formalism which allows to
formulate the problem on the lattice in $(2+1)$ dimensions. We discuss the
numerical complexity of the L\"uscher technique, systematic errors introduced
by polynomial approximation and introduce some improvements which reduce long
autocorrelations. In particular we show that preconditioning of the fermionic
matrix speeds up the algorithm and extends the available range of parameters.
We investigate the magnetic and the one-particle properties of the Hubbard
Model at half-filling and show that they are in qualitative agreement with the
existing Monte Carlo data and the mean-field predictions.
