Optimal quantum algorithms to simulate strongly correlated fermionic systems

We discuss quantum simulation of strongly correlated fermionic lattice models with nearest-neighbor interacting qubit arrays. We improve an existing quantum algorithm to prepare arbitrary Slater determinants by exploiting a symmetry in their representations. We present a quantum algorithm to prepare an arbitrary fermionic Gaussian state with O(N²) gates and O(N) circuit depth. This algorithm---unlike existing ones that rely on translational symmetry---is completely general and is useful to simulating disordered systems and quantum impurity models. These two algorithms are optimal because the numbers of gates equal to the numbers of parameters to describe the quantum states to be prepared. We also present an algorithm to implement the 2-dimensional (2D) fermionic Fourier transformation on a 2D qubit array with O(N^1.5) gates and O(N^1/2) circuit depth. Both scalings are optimal because they represent the minimum cost for quantum information to travel through the qubit array. We also present methods to simulate each time step in the evolution of the 2D Hubbard model---again on a 2D qubit array---with O(N) gates and O(N^1/2) circuit depth. Finally, we discuss the physical significance of these algorithms using the Fermi-Hubbard model as an example.