Quantum dot spin qubits are a promising platform for realizing quantum information technologies, which can theoretically perform calculations such as factoring large integers that are otherwise intractable using classical computing methods. However, quantum dot qubit technology is still in its developmental phases, with current experimental devices capable of holding only a few (less than 10) noisy qubits. Even with only a small number of quantum dots, interesting experiments can be performed, simulating physical systems and observing many-body phenomena which are otherwise difficult to study or numerically simulate classically.

In the first part of this thesis, we analytically examine valley states in Silicon, which is one obstacle which can potentially lead to information loss in Silicon qubits. Using a perturbative method, we calculate the dynamics of two exchange-coupled quantum dots in which there is a valley degree of freedom. We find that the spin states can become entangled with the valley states of the system if the electrons are not initialized to the correct valley states, which can adversely affect quantum computations performed on these systems. In the second part of this thesis, we detail how quantum dot plaquettes can simulate the Hubbard model and give many analytic results for different magnetic phenomena that arise under this model. These results include examples of Nagaoka ferromagnetism, violations of Hund's rule, and situations where flatband ferromagnetic ground states are necessarily degenerate with nonferromagnetic states. These phenomena all require only a few quantum dots, and are observable with current experimental technologies.