  Quantum computing is in an era of limited resources. Current hardware lacks
high fidelity gates, long coherence times, and the number of computational
units required to perform meaningful computation. Contemporary quantum devices
typically use a binary system, where each qubit exists in a superposition of
the $\ket{0}$ and $\ket{1}$ states. However, it is often possible to access the
$\ket{2}$ or even $\ket{3}$ states in the same physical unit by manipulating
the system in different ways. In this work, we consider automatically encoding
two qubits into one four-state qu\emph{quart} via a \emph{compression scheme}.
We use quantum optimal control to design efficient proof-of-concept gates that
fully replicate standard qubit computation on these encoded qubits.
  We extend qubit compilation schemes to efficiently route qubits on an
arbitrary mixed-radix system consisting of both qubits and ququarts, reducing
communication and minimizing excess circuit execution time introduced by
longer-duration ququart gates. In conjunction with these compilation
strategies, we introduce several methods to find beneficial compressions,
reducing circuit error due to computation and communication by up to 50\%.
These methods can increase the computational space available on a limited
near-term machine by up to 2x while maintaining circuit fidelity.
