Questions on Neural-Network Quantum States for Fermions

[zipeilee]

Hi,
I am a beginner in NNQS and VMC, and I have encountered some confusion while trying to construct NNQS for strongly correlated fermion problems.

As I understand it, the input configurations to the network correspond to the Fock basis. In other words, a many-body state can be expanded in the Fock basis (occupation number configurations) as

$$ |\Psi\rangle = \sum_\sigma C(\sigma)|\sigma\rangle, $$

and the neural network essentially learns the mapping $C(\sigma)$ (complex amplitude).

For both bosons and fermions, the Fock basis already encodes the symmetry or antisymmetry constraints of the wave function. That is, the input $\sigma$ already carries prior information about the required symmetry. Therefore, especially in the fermion case, one does not need to explicitly enforce antisymmetry in the network. From this perspective, we only need to promote the input and operators to the corresponding particle representation; the rest is essentially identical to NNQS for spin models and requires no modification.

For example, one could directly use the ViT tutorial to solve the many-body ground state of fermions, simply extending the input from $(0, 1)$ to $(0, \uparrow, \downarrow, \uparrow\downarrow)$, while using nk.hilbert.SpinOrbitalFermions and netket.operator.fermion to define the fermionic operators. After VMC optimization, the resulting state is the physical many-body ground state.

Some works, such as arXiv:2411.07144 and Nat Commun 16, 8464 (2025), indeed adopt this approach: the network directly accepts occupation number configurations, outputs complex amplitudes, and directly represents a many-body state. I will refer to this as case 1.

However, when reading the tutorial Lattice Fermions, from Slater Determinants to Neural Backflow Transformations, I noticed that the authors explicitly design the network to respect fermionic antisymmetry: the network only outputs single-particle states, which are then combined with an additional antisymmetrization step. This is understandable if the network is a mean-field Slater ansatz; but even in the neural backflow case, the network outputs single-particle states that still require antisymmetric combination. I will refer to this as case 2.

My questions are:

1. What is the fundamental difference between case 2 and case 1? Is explicitly considering fermionic antisymmetry theoretically necessary? Is case 1 incorrect?
2. I noticed that arXiv:2411.07144 and Nat Commun 16, 8464 (2025) use custom networks and libraries, not NetKet. Are nk.hilbert.SpinOrbitalFermions and netket.operator.fermion in NetKet also defined based on the Fock basis?
3. Both papers seem to use the Jordan-Wigner transformation to map fermions to non-local spin chains. Is this mapping the reason why their approach works?

Ultimately, my main concern is: if I want to directly apply a ViT model to fermion problems, following all steps in the ViT tutorial without adding any extra handling for antisymmetry, is this feasible? Will the resulting wave function be physical?

[gcarleo]

hi, good question. Both approches are perfectly valid. In the first case in fact you can also use other mappings that are different than the Jordan Wigner, ad we showed in this early paper https://www.nature.com/articles/s41467-020-15724-9 . The tutorial on fermions in NetKet discusses only the second approach because it's typically the more appropriate for periodic lattice Hamiltonians (not for molecules as in the paper above). Notice that there is no mistake in taking the determinant on second quantization, since the amplitudes of the free fermion state are exactly a determinant, even in second quantization.The backflow transformation is then typically easier and smoother to learn then the full mapping n->psi(n) that instead you do for approach A. In any case, it is an open research question whether approach A can be made work at the same level of precision than B on periodic lattice Hamiltonians.

[zipeilee]

Thank you very much for your detailed explanation!

If I understand correctly, this means that using a neural network architecture to directly learn lattice fermion systems such as the Hubbard model (i.e., approach A) is still worth exploring, right?

I also have a few follow-up questions:

Why is approach B (the Slater/backflow type) typically preferred for lattice fermion systems?
What makes lattice models more challenging than electronic structure problems treated from ab-initio?

In approach A, are mappings such as the Jordan–Wigner or Bravyi–Kitaev transformations strictly necessary?
Why do we need to map the fermionic occupation basis to spin chains instead of directly working in the fermionic Fock-space representation?

Thank you again for your time and insight!