  The interaction between electrons in arrays of electrostatically defined
quantum dots is naturally described by a Fermi-Hubbard Hamiltonian. Moreover,
the high degree of tunability of these systems make them a powerful platform to
simulate different regimes of the Hubbard model. However, most quantum dot
array implementations have been limited to one-dimensional linear arrays. In
this letter, we present a square lattice unit cell of 2$\times$2 quantum dots
defined electrostatically in a AlGaAs/GaAs heterostructure using a double-layer
gate technique. We probe the properties of the array using nearby quantum dots
operated as charge sensors. We show that we can deterministically and
dynamically control the charge occupation in each quantum dot in the single- to
few-electron regime. Additionally, we achieve simultaneous individual control
of the nearest-neighbor tunnel couplings over a range 0-40~$\mu$eV. Finally, we
demonstrate fast ($\sim 1$~$\mu$s) single-shot readout of the spin state of
electrons in the dots, through spin-to-charge conversion via Pauli spin
blockade. These advances pave the way to analog quantum simulations in two
dimensions, not previously accessible in quantum dot systems.
