Colloquium: Qudits for decomposing multiqubit gates and realizing quantum algorithms

Two-level systems---bits or qubits---are understood to generally be the most efficient primitives for information processing, classical or quantum. But this is not to say that there are no roles to be played by multilevel systems. This Colloquium surveys these possible roles for the quantum case. Here we speak of qudits: d-level quantum systems. In one interesting example, the use of just one three-level system permits a drastic simplification of the ``Toffoli gate, '' the basic three-qubit primitive of reversible logic. A survey is given of various qudit-qubit embeddings, and the current state of quantum computing experiments using qudits is reviewed.