# What operators preserve antisymmetry?

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 $$$$f_{swap} = 1 + a_p^\dagger a_q + a_q^\dagger a_p - a_p^\dagger a_p - a_q^\dagger a_q.$$$$ I am, for the moment, only interested in one-body electronic Hamiltonians, such as the kinetic or external potential terms. This one-body Hamiltonians have the form $$$$H = h_{pq} a_p^\dagger a_q + \text{Hermitian conjugate}$$$$ However, if I try to perform the commutator of $$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?

• Is $h_{pq}$ real? Feb 27, 2022 at 0:20
• If so, then $H$ and $f$ should certainly commute! Feb 27, 2022 at 0:21
• Thanks @JahanClaes. Yes, I had taken $h_{pq}$ real. But if I try to commute them (see the collab above) it doesn't seem to. That's why I am surprised. Feb 27, 2022 at 10:14