  We present an architecture for encoding two qubits within the optical "clock"
transition and nuclear spin-1/2 degree of freedom of neutral ytterbium-171
atoms. Inspired by recent high-fidelity control of all pairs of states within
this four-dimensional ququart space, we present a toolbox for intra-ququart
(single atom) one- and two-qubit gates, inter-ququart (two atom) Rydberg-based
two- and four-qubit gates, and quantum nondemolition (QND) readout. We then use
this toolbox to demonstrate the advantages of the ququart encoding for
entanglement distillation and quantum error correction which exhibit superior
hardware efficiency and better performance in some cases since fewer two-atom
(Rydberg-based) operations are required. Finally, leveraging single-state QND
readout in our ququart encoding, we present a unique approach to studying
interactive circuits as well as to realizing a symmetry protected topological
phase of a spin-1 chain with a shallow, constant-depth circuit. These
applications are all within reach of recent experiments with neutral
ytterbium-171 atom arrays or with several trapped ion species.
