I have just started to read about DMRG and MPS.
It is said that in case of simple 1D chain with spins states $|\uparrow\rangle$; $|\downarrow\rangle$ and any state in the complete Hilbert space of such a system could be written as :
Where each index runs over the local basis at each site. This presents a complexity of the order $2^N$ and by writing it in the form of matrix product states we would making a simplification a mean field simplification and reduce the complexity to $2N$.
A General case:
Suppose I have Heisenberg chain kind of a 1D Lattice problem with N sites, where each site lives on a Hilbert space of dimensionality D. Given the fact that the dimensionality of Hilbert space of the entire system scales exponentially as $D^N$ because of the entanglement, it becomes a very complex problem to solve.
- How does writing the system in terms of MPS reduce the complexity of the problem and bring it down to $ND$ ?
- What is the link between using MPS ansatz and mean field approximation?