Variational simulations of fermionic matter with neural-network quantum states

Neural networks have proven to form a powerful variational representation for the quantum states of quantum many-body systems, especially in >1d. While most research has focused on studying quantum spin systems, extending Neural-network Quantum states (NQS) to fermionic degrees of freedom remains underexplored. The main reason is the additional challenges introduced by fermionic anti-commutation relations. A first design choice in variational simulations of fermionic systems is to represent the state in first or second quantization. While the first requires explicit anti-symmetrization of the wave function through the use of determinants, the second can introduce non-local interactions in >1d after Jordan-Wigner transformations. In this work, we discuss our recent progress on how to obtain the low-energy spectrum of fermionic Hamiltonians (on a lattice), by mapping local fermionic Hamiltonians onto local spin Hamiltonians, as well as embedding symmetries in fermionic NQS. We will demonstrate how this allows us to use the power of NQS (originally designed for quantum spin systems) to study fermionic matter with Variational Monte Carlo.