  Gate-based universal quantum computation is formulated in terms of two types
of operations: local single-qubit gates, which are typically easily
implementable, and two-qubit entangling gates, whose faithful implementation
remains one of the major experimental challenges since it requires controlled
interactions between individual systems. To make the most of quantum hardware
it is crucial to process information in the most efficient way. One promising
avenue is to use higher-dimensional systems, qudits, as the fundamental units
of quantum information, in order to replace a fraction of the qubit-entangling
gates with qudit-local gates. Here, we show how the complexity of multi-qubit
circuits can be lowered significantly by employing qudit encodings, which we
quantify by considering exemplary circuits with exactly known (multi-qubit)
gate complexity. We discuss general principles for circuit compression, derive
upper and lower bounds on the achievable advantage, and highlight the key role
played by entanglement and the available gate set. Explicit experimental
schemes for photonic as well as for trapped-ion implementations are provided
and demonstrate a significant expected gain in circuit performance for both
platforms.
