Hi admin,
This is probably a standard question with some literature reference and answer but my question here is:
Given an MPS $$| \psi \rangle$$, say, I want to construct a bipartite reduced density matrix (where I trace out either left half or right half of a 1D MPS wavefunction to obtain $$\rho_L$$ or $$\rho_R$$ depending on which half I trace out). My algorithm works as follows:
(1) Outer product of $$| \psi \rangle$$ with itself.
(2) Say, I want to keep left half and trace out right half, I cast my MPS into mixed canonical form with orthogonality center at the middle of a spin chain MPS.
(3) I the trace out by contracting the legs of the density matrix MPO (from outer product of MPS) starting from the right end of the MPS successively up to the orthogonality center.
My question is: what is the complexity of the algorithm above in constructing the MPO reduced density matrix above, in terms of, say, bond dimension $$D$$, spin chain length $$L$$ (we considered bipartition so reduced density matrix still has length $$L/2$$), exam spin with $$d$$ dimensions (for simplicity, say take spin-1/2 Ising chain so $$d = 2$$), and other relevant variables?