Improve implementation of MPO-MPO multiplication

[20akshay00]

The current MPO * MPO implementation performs site-wise multiplication followed by a global changebonds truncation. This approach has the issue that it pushes all the norm to one tensor at the end. Alternatively, one may perform a sequential QR in a left-right pass while multiplying the MPOs to maintain canonical form and then truncate the bonds on the right-left pass.

[VictorVanthilt]

You can do this in the same way as applying an MPO to an mps, by converting one of the mpos to an mps and applying the other on this mps. Using finite VOMPS (called DMRG() in mpskit) for finite systems or VOMPS() for infinite systems is often a good choice.

# mpo1 and mpo2 are mpos
# i want to make mpo2 * mpo1
mpo1_mps = convert(FiniteMPS, mpo1) # bends top leg down (it becomes the second physical leg)


answer, _ = approximate(mpo1_mps, (mpo2, mpo1_mps), my_favorite_algorithm()) # applies mpo2 to the first physical leg of each tensor 
answer = convert(FiniteMPO, answer)

[lkdvos]

wew, I had no clue this actually just works out of the gate, that's kind of funny :p

[20akshay00]

This still seems to accumulate the norm onto one tensor, but I'm not sure at what point it happens.

[lkdvos]

It's the same point about the gauging of the MPS, which in this case is implicit. The norm of the MPO is the norm of the MPS is the norm of the center gauge tensor, which is why it explodes.
By the way, thinking about this a bit more, can you renormalize your operator such that the norm is not size-extensive?