Assume, I'm using a system of qubits to simulate a fermionic system.

If I'm using the second-quantized formalism (e.g. orbitals in quantum chemistry), the anti-symmetric nature of the fermionic wave functions can be taken into account by means of the Jordan-Wigner or Brayi-Kitaev transformation. Roughly speaking, this gives a 1-to-1 correspondence between the physical (fermionic) and logical (bosonic) DOFs, with operators of certain locality. So, in terms of qubits resources, this is as efficient as can be.

Now, could anyone please elaborate how the issue of the wave function anti-symmetry is typically resolved within the first-quantized approach? (In other words, when doing lattice simulations.)

  • $\begingroup$ Are you asking, how to get qubits such that they respect the exchange rule, written here for two arbitrary electrons $e_ie_j=-e_je_i$? $\endgroup$
    – draks ...
    Commented Apr 27, 2020 at 10:25
  • $\begingroup$ Well, in the first-quantized formalism (i.e. when one dicretizes the space, and then at the each point of the spacial lattice discretizes the function values) it is not necessarily clear what is an 'electron'. But in a certain sense - yes. $\endgroup$
    – mavzolej
    Commented Apr 27, 2020 at 16:24
  • $\begingroup$ Are exchanges restricted by the lattice or is any fermion exchangable with every other? Sorry I'm not an expert on lattices... $\endgroup$
    – draks ...
    Commented Apr 27, 2020 at 20:01
  • $\begingroup$ Not sure about lattices, but in first quantization antisymmetrization is directly enforced by using Slater determinants $\endgroup$
    – chrysaor4
    Commented Nov 25, 2021 at 14:10
  • 1
    $\begingroup$ I'm pretty sure that in first-quantized simulations, you just initialize your wavefunction in a state that happens to be antisymmetric, and then you make sure all subsequent operations preserve the symmetric/antisymmetric subspaces. $\endgroup$ Commented Mar 6, 2022 at 3:54


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