Periodic MPS / SBS machines

Created by: stavros11

Hello everyone,

Trying not to increase the volume of pulls with unfinished code, I was planning to propose this pull after having everything working a bit better and well optimized. Given the current timeline, it might be a good idea to share earlier so that you also have some time to review the progress and decide whether this contribution is going to be useful or how we can proceed.

The uploaded files are the following:

• mps_periodic.hpp: Implements a periodic MPS machine where all matrices are square with the same (bond) dimension given by user. It is possible to set translational symmetry (same set of matrices repeated across the chain).
• mps_diagonal.hpp: Implements a machine identical to mps_periodic.hpp where all matrices are diagonal (using cwiseProduct instead of matrix multiplication).
• abstract_mps.hpp: Contains virtual methods so that the above classes can be used for SBS calculations.
• sbs.hpp: Implements a SBS (String Bond State = Product of MPS strings) machine. All SBS strings are periodic.

And a README explaining the JSON settings.

WARNING: Unit test failures

• The real test of DerLog for all machines. The imaginary part has no problems and the machines seem to converge to the correct energies for small (N<10) 1D Ising and Heisenberg Hamiltonians. I am not sure where the unit test error comes from - in my computer the Jastrow machines fail the same test too.
• I use exceptions in the SBS machine which I think are not allowed in the Google style. This is only in the from_json function and does not affect calculations, so it will be replaced soon (sorry for that).

Some additional notes:

1. The look up tables store all left and right contractions. This is efficient in the one-flip case but might not be the optimal when more than one spins are flipped. Storing middle contractions could assist in computations for large systems but would add time in the look up update. It is not straightforward (to me) what is the best to do. Optimizing this is among my priorities.
2. All contractions are calculated from scratch every time DerLog is called. Using look up tables in DerLog would help this and other machines, but this requires small changes in the VMC part (and possibly some other parts too).
3. User can explicitly assign which sites go to each string in SBS through the JSON. This allows the greatest flexibility but might be a bit hard/non-automatic from the user's perspective and also allows non-physical cases (eg. strings that jump sites). An easy way to bypass this is by writing a simple Python wrapper that creates the "sites to strings" lists for the JSON. A more formal way could be to use the graph object directly in C++. I am not yet sure what's the best way to simplify things while still keeping the settings quite general.
4. The "sites to strings" option in SBS can be used to generalize MPS too (1-string SBS = MPS). Hence the SBS machine can be used to get different MPS shapes (eg. snake MPS), which is not supported in my pure MPS implementations.
5. Weight loading does not work. Machines initialize randomly after loading.
6. My machines are possibly slower than all current machines and the number of parameters is typically larger (eg. ~N^3 for MPS vs ~N^2 for RBM). In any case, I am far from C++ expert, so please comment if you find any "newbie" errors in my code. It seems that the diagonal version is much faster even for small systems, which might be just because the number of parameters is smaller or because the contractions are inefficient.
7. I have an implementation for open boundary MPS which allows different matrix dimensions across the chain. If there is interest I add that one too, probably after I manage to add a feature to transform it to its canonical form.

Thank you very much and I hope this is useful. Many thanks to I. Glasser for getting me involved and explaining many things related to these kind of states. Note that the code hasn’t been reviewed, therefore I am responsible for any issues!

Stavros