  The quantum many-body problem is an important topic in condensed matter
physics. To efficiently solve the problem, several methods have been developped
to improve the representation ability of wave-functions.
  For the Fermi-Hubbard model under periodic boundary conditions, current
state-of-the-art methods are neural network backflows and the hidden fermion
Slater determinant.
  The backflow correction is an efficient way to improve the Slater determinant
of free-particles.
  In this work we propose a tensor representation of the backflow corrected
wave-function, we show that for the spinless $t$-$V$ model, the energy
precision is competitive or even lower than current state-of-the-art fermionic
tensor network methods.
  For models with spin, we further improve the representation ability by
considering backflows on fictitious particles with different spins, thus
naturally introducing non-zero backflow corrections when the orbital and the
particle have opposite spins.
  We benchmark our method on molecules under STO-3G basis and the Fermi-Hubbard
model with periodic and cylindrical boudary conditions.
  We show that the tensor representation of backflow corrections achieves
competitive or even lower energy results than current state-of-the-art neural
network methods.
