Proposal for slight modification to function `generate_hamiltonian`

[michaelkaicher]

Hi,

when playing with some active space Hamiltonians with pyscf and generating them as FermionOperator objects (or rather InteractionOperator objects), I found that the resulting Hamiltonian included a lot of unnecessary terms: namely, operators such as

h_{kklm} a_k^dag a_k^dag a_l a_m,

where h_{kklm} was different than zero and the operator thus included (but the operator a_k^dag a_k^dag a_l a_m is zero, so one should just set h_{kklm} to zero right away (k,l,m are here spinorbital indices).

Maybe this problem could be solved by just adding:
if not (p==q or r==s):
before
# Same spin two_body_coefficients[2 * p, 2 * q, 2 * r, 2 * s] = (two_body_integrals[p, q, r, s] / 2.) two_body_coefficients[2 * p + 1, 2 * q + 1, 2 * r + 1, 2 * s + 1] = (two_body_integrals[p, q, r, s] / 2.)
in line 272 of https://github.com/quantumlib/OpenFermion/blob/v1.6.0/src/openfermion/hamiltonians/hartree_fock.py#L245-L291

Would this cause any problems?

Cheers,
Michael.

[ANUKOOL324]

allow me to look on this issue @fdmalone

[matti-sarjala]

A useful way to frame the fix may be to distinguish tensor coefficient storage from the actual fermionic operator algebra.

Even if a two-body tensor entry is nonzero, the corresponding normal-ordered term can still vanish algebraically when it contains repeated creation or repeated annihilation operators on the same spin orbital, e.g.

a_k^\dagger a_k^\dagger a_l a_m = 0

or

a_k^\dagger a_l^\dagger a_m a_m = 0.

Possible acceptance criteria:

1. Terms with repeated creation indices or repeated annihilation indices should not appear in the generated FermionOperator.
2. The generated Hamiltonian should be operator-equivalent to the previous result, but with fewer zero terms.
3. Add a small regression test where a same-spin two-body coefficient would previously generate a term with repeated fermionic operators.
4. Check that this does not remove valid opposite-spin terms, where the spin-orbital indices are different even if they originate from the same spatial orbital.

That would make the invariant explicit: the tensor may contain entries from the integral construction, but the emitted fermionic operator should respect the nilpotency of fermionic creation and annihilation operators.