[Feature Request] `left_orth` and `right_orth` without calculating Q

[Yue-Zhengyuan]

I would like to see an option to make left_orth and right_orth to calculate only R and L without calculating Q (similar to scipy.linalg.qr with option mode = 'r').

An application where Q is not needed is putting an open boundary MPS into canonical form.

[Jutho]

I am confused why you don't need Q if you are transforming an OBC MPS in canonical form? I think, if you have a very rectangular matrix, the fastest way of getting L R is by doing the Cholesky decomposition of A'*A, but you might loose a little bit of precision if there are small singular values.

[Yue-Zhengyuan]

Sorry I may have used "canonical form" in an unconventional way. I'm actually referring to the "symmetric gauge" of the MPS, where each MPS tensors are obtained from the "Vidal gauge" by absorbing square root of the bond weights (using the language of SciPostPhys.15.6.222).

The conversion from an arbitrary OBC MPS to the symmetric gauge is described in the opening docstring of the 3-site simple update in PEPSKit. There I only need the R, L matrices to construct the "projectors" on each virtual bond. The MPS tensors are brought to the symmetric gauge after being acted on by these projectors.

Nevertheless, in this particular application, the reshaped matrix to be fed into QR/LQ is indeed quite rectangular.