ternary tensor contraction

[janbrandejs]

Q-Chem (specifically Jonas Leitner developing ADC/ADCC using libint) developers at ReLibQC25 reported the need for a ternary tensor contraction to correctly encapsulate the bottleneck tensor operation in their code while allowing performance optimization on the tensor lib side.

D += A*B*C

[pauldj]

One or more of the tensors are sparse, correct?

[evaleev]

At best block sparse in their app.

[LinusBDittmer]

One or more of the tensors are sparse, correct?

The application for ternary contractions I am thinking of involve the on-the-fly construction and immediate contraction of tensor intermediates, e. g. the MP amplitudes. Since the MP amplitudes in canonical orbitals are always calculated as some tensor contraction followed by an element-wise multiplication with an orbital energy denominator, implicit construction and contraction always involves a ternary contraction. To my best knowledge, the adcman and adcc code currently do not make use of this and instead choose to simply build the amplitudes and subsequently contract them, however these ternary contractions are quite helpful in lower order MP and ADC to lower memory requirements.

To extend @evaleev's argument, possible ternary contractions arise from Kronecker Deltas and therefore already remove almost all useable sparsity. Since we use canonical orbitals, we are only left with sparsity from spatial symmetry and spin blocks, which makes the relevant tensors at best block sparse.

[jonasleitner]

As far as I understand, there are 2 different types of ternary contractions one could encounter:

• $d = \sum_i A_i B_i C_i$ : the simultaneous contraction of 3 tensors, e.g., on-the-fly evaluation of MP amplitudes as mentioned by @LinusBDittmer.
• $\sum_{Pkc} B^P_{jk} B^P_{bc} Y^{ac}_{ik}$ : an actual example from the RI-ADC implementation that I found in our codebase. This operation can be performed as 2 subsequent binary contractions (first contract over $P$, then over $k$ and $c$). However, the contraction over P would lead to the reconstruction of the full ERI tensor ($B$ are the symmetrically RI-decomposed ERIs). Therefore, we use a ternary contraction that essentially enables a block wise lazy evaluation of the full ERI tensor and thereby avoids its full construction.

One or more of the tensors are sparse, correct?

Essentially all three tensors should at best be block sparse due to their (point-group, permutational, spin) symmetry.