Thresholding in QHAT

[reuben-tate]

There are some discrepancies in Hamiltonian term count for molecular Hamiltonians that only difference in interatomic distances. Theoretically, they should have the same number of terms, but I believe that thresholding is occurring that eliminates terms whose coefficients are considered too small. I believe the thresholding is happening in OpenFermion. Consider the following function call sequence:

apply_active_space [in hamiltonian_generator/hamgen.py of QHAT]
-> get_molecular_hamiltonian [in openfermion/chem/molecular_data.py]
-> spinorb_from_spatial [in openfermion/chem/molecular_data.py]

In the last function, there are the following two lines which seem to indicate thresholding:

# Truncate.
    one_body_coefficients[numpy.absolute(one_body_coefficients) < EQ_TOLERANCE] = 0.0
    two_body_coefficients[numpy.absolute(two_body_coefficients) < EQ_TOLERANCE] = 0.0

In openfermion/config.py the EQ_TOLERANCE is set to 10^(-8) by default. There could be other places where thresholding/truncation is also happening, I have not checked thoroughly.

I am not necessarily suggesting a specific code change or fix, but I think this is something that should be considered and we should be intentional regarding our decisions on this. Some specific things to consider:

• Is the thresholding really the issue that is causing the discrepancy mentioned in the first paragraph?
• Which parts of the QHAT script result in some thresholding taking place?
• Is it possible to modify QHAT to remove thresholding entirely or to adjust it?
• Does removing the thresholding result in the expected number of Hamiltonian terms?
• Do we have an easy way of knowing how many Hamiltonian terms there should be (without thresholding) for testing purposes?

[BrendanKKrueger]

I think we expect that there will be less terms as the interatomic spacing increases, particularly due to truncation exactly as you identified. There are some quick-and-dirty hacks we could try just to test things out:

• Import EQ_TOLERANCE and set it to zero before calling get_molecular_hamiltonian
• Copy molecular_data.py, remove those two lines, and call our modified version instead of the one from OpenFermion.

Both of these could have unintended side effects, which is why I call them "quick-and-dirty hacks". But they might quickly get you to the kind of answer that you need and/or point to more robust solutions.

[AlbertLee125]

• The thresholding occurs when QHAT calls OpenFermion’s get_molecular_hamiltonian().
• OpenFermion’s spinorb_from_spatial() applies a fixed ($$10^{-8}$$) cutoff and overwrites smaller nonzero coefficients with zero.
• For Li(_2), this removed 36 one-body and 2,064 two-body tensor entries.
• The main issue is that QHAT currently has no way to configure or disable this upstream threshold.
• We should expose the threshold in QHAT and compare ground-state energies at ($$10^{-8}$$) and ($$0$$).