Why Matrix Product State (MPS) representation provide fast computation? I am interested in tensor networks and I am trying to understand why MPS (for example) provide an efficient representation of a quantum state. In order to transform the quantum state in MPS representation you have to apply repetitively the Singular-Value-Decomposition (SVD), but you have to initially start with the big vector and then apply the SVDs in the correct way. If you are able to start with this state why do we care with MPS?
 A: Typically one does not start with the full many-body state of interest and compress down to an MPS. Usually the reason to use MPS methods is because we don't have direct access to the full state. For example, if you're interested in the ground state of a complicated 1d Hamiltonian, then the idea is to use an algorithm like DMRG to find an approximate MPS representation of the ground state. In this procedure you never start with the full many-body ground state and compress it, rather you start with a simple state and variationally optimize to achieve the lowest possible energy MPS.
One reason you might be confused is because many reviews on MPS methods start with a full many-body state $|\psi \rangle = \sum_{\{s\}} c_{s_1 \ldots s_N} |s_1 \ldots s_N \rangle$ and then demonstrate how one can write $|\psi \rangle$ as an MPS by performing repeated SVDs on $c_{s_1 \ldots s_N}$. But the point of this exercise is to show that any state can in principle be written as an MPS. But this is not what one actually does in practice, when $c_{s_1 \ldots s_N}$ is exponentially complex.
