Quantum encodings in spin systems and harmonic oscillators

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.