I would like to understand better what kind of operators maintain antisymmetry as explained in Quantum simulation of chemistry with sublinear scaling in basis size:
Evolution under the Hamiltonian will maintain antisymmetry provided that it exists in the initial state (a consequence of fermionic Hamiltonians commuting with the electron permutation operator).
I have further assumed that the electron permutation operator is given by Low-Depth Quantum Simulation of Materials \begin{equation} f_{swap} = 1 + a_p^\dagger a_q + a_q^\dagger a_p - a_p^\dagger a_p - a_q^\dagger a_q. \end{equation} I am, for the moment, only interested in one-body electronic Hamiltonians, such as the kinetic or external potential terms. This one-body Hamiltonians have the form \begin{equation} H = h_{pq} a_p^\dagger a_q + \text{Hermitian conjugate} \end{equation} However, if I try to perform the commutator of $f_{swap}$ and $H$ above using OpenFermion, it does not return 0: Collab notebook. I have also tried with Givens rotations $e^{\theta(a_p^\dagger a_q- a_q^\dagger a_p)}$ and it does not seem to work either.
Is the problem that $f_{swap}$ is not the electronic permutation operator? Or what am I doing wrong?
