  We investigate the application of matrix product states to the Hubbard model
in one spatial dimension with both of open and periodic boundary conditions. We
develop the variatinal method that the optimization of the variational
parameters is carried out locally and sequentially in the framework of matrix
product operators (MPO) by including the sign, due to the anti-commutation
relation of fermion operators, in the matrix element of MPO. The numerical
accuracy of the ground state energy is examined.
