Quantum simulations of ultracold atoms in fermionic phase space

High-temperature superconductivity is a promising, energy-saving technology with the potential for eliminating power transmission energy loss. The Hubbard model is a widely used although the simplified model for high- temperature superconductivity. One promising and exact method of solving Fermi systems involves fermionic phase space representations. This thesis derives a range of Majorana identities and utilizing the fermionic Q-function and the P-functions to study the time evolution of Fermi systems. Shock wave formation and its dynamics, Majorana Hubbard model are also treated by keeping in mind the 2D Hubbard model as our long term goal.