  The paradigm behind digital quantum computing inherits the idea of using
binary information processing. The nature in fact gives much more rich
structures of physical objects that can be used for encoding information, which
is especially interesting in the quantum mechanical domain. In this Colloquium,
we review several ideas indicating how multilevel quantum systems, also known
as qudits, can be used for efficient realization of quantum algorithms, which
are represented via standard qubit circuits. We focus on techniques of
leveraging qudits for simplifying decomposition of multiqubit gates, and for
compressing quantum information by encoding multiple qubits in a single qudit.
As we discuss, these approaches can be efficiently combined. This allows
reducing in the number of entangling (two-body) operations and the number of
the used quantum information carriers compared to straightforward qubit
realizations. These theoretical schemes can be implemented with quantum
computing platforms of various nature, such as trapped ions, neutral atoms,
superconducting junctions, and quantum light. We conclude with summarizing a
set of open problems, whose resolving is an important further step towards
employing universal qudit-based processors for running qubit algorithms.
