  Efficient scaling and flexible control are key aspects of useful quantum
computing hardware. Spins in semiconductors combine quantum information
processing with electrons, holes or nuclei, control with electric or magnetic
fields, and scalable coupling via exchange or dipole interaction. However,
accessing large Hilbert space dimensions has remained challenging, due to the
short-distance nature of the interactions. Here, we present an atom-based
semiconductor platform where a 16-dimensional Hilbert space is built by the
combined electron-nuclear states of a single antimony donor in silicon. We
demonstrate the ability to navigate this large Hilbert space using both
electric and magnetic fields, with gate fidelity exceeding 99.8% on the nuclear
spin, and unveil fine details of the system Hamiltonian and its susceptibility
to control and noise fields. These results establish high-spin donors as a rich
platform for practical quantum information and to explore quantum foundations.
