  Simulating strongly correlated fermionic systems is notoriously hard on
classical computers. An alternative approach, as proposed by Feynman, is to use
a quantum computer. Here, we discuss quantum simulation of strongly correlated
fermionic systems. We focus specifically on 2D and linear geometry with nearest
neighbor qubit-qubit couplings, typical for superconducting transmon qubit
arrays. We improve an existing algorithm to prepare an arbitrary Slater
determinant by exploiting a unitary symmetry. We also present a quantum
algorithm to prepare an arbitrary fermionic Gaussian state with $O(N^2)$ gates
and $O(N)$ circuit depth. Both algorithms are optimal in the sense that the
numbers of parameters in the quantum circuits are equal to those to describe
the quantum states. Furthermore, we propose an algorithm to implement the
2-dimensional (2D) fermionic Fourier transformation on a 2D qubit array with
only $O(N^{1.5})$ gates and $O(\sqrt N)$ circuit depth, which is the minimum
depth required for quantum information to travel across the qubit array. We
also present methods to simulate each time step in the evolution of the 2D
Fermi-Hubbard model---again on a 2D qubit array---with $O(N)$ gates and
$O(\sqrt N)$ circuit depth. Finally, we discuss how these algorithms can be
used to determine the ground state properties and phase diagrams of strongly
correlated quantum systems using the Hubbard model as an example.

Quantum Physics