  Low depth measurement-based quantum computation with qudits ($d$-level
systems) is investigated and a precise relationship between this powerful model
and qudit quantum circuits is derived in terms of computational depth and size
complexity. To facilitate this investigation a qudit `unbounded fan-out'
circuit model, in which a qudit may be quantum-copied into an arbitrary number
of ancillas in a single time-step, is introduced and shown to be capable of
implementing interesting $n$-qudit unitaries in constant depth. A procedure for
reducing the quantum computational depth in the measurement-based model is then
proposed and using this it is then shown that there is a logarithmic depth
separation between the depth complexity of qudit measurement-based computation
and circuits composed of gates act on a bounded number of qudits. The
relationship is made precise by showing that the depth complexity of the qudit
measurement-based model is exactly equivalent to that of unbounded fan-out
circuits. These results illustrate that the well-known advantages inherent in
qubit measurement-based quantum computation are also applicable to the
higher-dimensional generalisation. As qudits are both naturally available and
have been shown to provide fundamental advantages over binary logic encodings,
this then suggests that the qudit measurement-based model is a particularly
appealing paradigm for universal quantum computation.
