ENH: Symmetries and parallelization support of einsum

[fellideal]

Proposed new feature or change:

Einsum has been fantastic for computing tensor contractions. However, there are two additional supports I believe would greatly enhance its functionality and efficiency.

(1) Symmetries:
Let's consider a tensor contraction such as A[a,b,c,d,e] * B[d,e,f,g,h] = C[a,b,c,f,g,h]. If we have symmetries such as A[a,b,c,d,e] = A[b,a,c,d,e], or A[a,b,c,d,e]=A[a,b,c,e,d], being able to utilize these symmetries could significantly enhance computational efficiency.

A package called ctf provides some support for this, but it seems to be only for symmetric/antisymmetric for adjacent indices. The link below leads to its GitHub repository:

https://github.com/cyclops-community/ctf

Further details can be found in the following documentation:

https://solomon2.web.engr.illinois.edu/ctf_python/ctf.html#module-ctf.core

However, from what I've seen, the efficiency isn't as good as one might hope (as noted in the following GitHub issue):

cyclops-community/ctf#136

(2) Parallelization:
Although it seems that tensordot supports parallelization
https://stackoverflow.com/questions/23650449/numpy-np-einsum-array-multiplication-using-multiple-cores

, when exporting MKL/OpenBLAS/OpenMP together with einsum for general contractions, the actual speed up is not quite realized, despite an increase in CPU load.

For binary tensor contraction, there are some packages, such as:

https://github.com/jackkamm/einsum2

However, for more general applications (more than two), it seems we're still lacking comprehensive solutions.

If these two features could be incorporated - symmetries and parallelization - into einsum, I believe it would greatly enhance its functionality and computational efficiency.

[priyank-p0]

@seberg I was looking up at older related threads and came across #14332. Since numpy doesn't natively support parallalization, apart from its integration with BLAS. I was thinking along the lines of solution provided in #14332 which calls for an integration with BLAS for contractions which rely on c_einsum backend call. Do you have any suggestions? Thank you !