  We show that higher-dimensional versions of qubits, or qudits, can be encoded
into spin systems and into harmonic oscillators, yielding important advantages
for quantum computation. Whereas qubit-based quantum computation is adequate
for analyses of quantum vs classical computation, in practice qubits are often
realized in higher-dimensional systems by truncating all but two levels,
thereby reducing the size of the precious Hilbert space. We develop natural
qudit gates for universal quantum computation, and exploit the entire
accessible Hilbert space. Mathematically, we give representations of the
generalized Pauli group for qudits in coupled spin systems and harmonic
oscillators, and include analyses of the qubit and the infinite-dimensional
limits.
