  The computation of the ground state (i.e. the eigenvector related to the
smallest eigenvalue) is an important task in the simulation of quantum
many-body systems. As the dimension of the underlying vector space grows
exponentially in the number of particles, one has to consider appropriate
subsets promising both convenient approximation properties and efficient
computations. The variational ansatz for this numerical approach leads to the
minimization of the Rayleigh quotient. The Alternating Least Squares technique
is then applied to break down the eigenvector computation to problems of
appropriate size, which can be solved by classical methods. Efficient
computations require fast computation of the matrix-vector product and of the
inner product of two decomposed vectors. To this end, both appropriate
representations of vectors and efficient contraction schemes are needed.
  Here approaches from many-body quantum physics for one-dimensional and
two-dimensional systems (Matrix Product States and Projected Entangled Pair
States) are treated mathematically in terms of tensors. We give the definition
of these concepts, bring some results concerning uniqueness and numerical
stability and show how computations can be executed efficiently within these
concepts. Based on this overview we present some modifications and
generalizations of these concepts and show that they still allow efficient
computations such as applicable contraction schemes. In this context we
consider the minimization of the Rayleigh quotient in terms of the {\sc
parafac} (CP) formalism, where we also allow different tensor partitions. This
approach makes use of efficient contraction schemes for the calculation of
inner products in a way that can easily be extended to the mps format but also
to higher dimensional problems.
