Computational complexity of applying a series of MPO on a reduced density matrix

[Brian-Khor]

This question is related to my previous thread/question:

Computational complexity of constructing the reduced density matrix of an MPS

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 hal…

Given a reduced density matrix $$\rho_L$$ of size $$L/2$$ over a spin chain (traced out the right half of the spin chain from outer product of MPS), now I want to successively apply MPO $$e^{i \sigma_i^z \sigma_{i+1}^z}$$ for $$i= 1, \dots, L/2 -1$$ to the reduced density matrix MPO $$\rho_L$$, so I construct a for loop of the “ITensors.apply” method from the documentation below (under “Algebra Operations”, for the one with 2 MPOs) to apply the MPO from $$i = 1, \dots, L/2 -1$$ :

https://itensor.github.io/ITensors.jl/stable/MPSandMPO.html

What is the computational complexity of such operations, on relevant parameters? (In this case I’m not entirely sure of which parameters are relevant, like bond dimension of $$\rho_L$$, dimension of the two-site operators, etc?)