Tese de Mestrado Integrado, Engenharia Física, 2022, Universidade de Lisboa, Faculdade de CiênciasMany-body quantum problems are still very complex to solve and today’s solutions do not take
into account that the objects of the systems are entangled with each other. By using tensor
networks, we are able to describe the same system using drastically fewer coefficients. Instead
of 2N parameters, in certain systems, we only require N2m3
. In the Matrix Product State
(MPS) algorithm, it is included the Singular Value Decomposition (SVD), which allows us to
truncate the tensors and keep only the crucial information about the system, and the Density
Matrix Renormalization Group (DMRG), which allows us to obtain the lowest energy MPS wave
function of our system. The goal was to study the optimal value for the bond dimension when
renormalizing the scale of a complex network. The connectivity of each node of a Barab´asi
network was used as the input for the algorithm and the MPS was trained based on whether
the nodes share a connection or not. When measuring each site after training, we obtained two
different outputs corresponding to the two linear independent vectors that form the space of the
MPS. The MPS was trained using different values for the bond dimension. Nevertheless, only
the values equal to 2 and 3 produced viable results since it did not converge for different values
of the bond dimension. The connectivity of a Barab´asi network follows a power law proportional
to x
−p
. When the Barab´asi network was characterized we obtained that its connectivity followed
a power law proportional to x
−2.7
, which is between the theoretical values of 2 and 3. The value
obtained for p = 2.7 further proves the value obtained for the bond dimension between 2 and
3, meaning that the dimension of the space needed to fully describe the system is between said
values. Finally, two different methods are suggested which obtain the same results in a simpler
and quicker way
