Quantum computation promises to solve fundamental, yet otherwise intractable, problems across a range of active fields of research. Recently, universal quantum logic-gate sets-the elemental building blocks for a quantum computer-have been demonstrated in several physical architectures. A serious obstacle to a full-scale implementation is the large number of these gates required to build even small quantum circuits. Here, we present and demonstrate a general technique that harnesses multi-level information carriers to significantly reduce this number, enabling the construction of key quantum circuits with existing technology. We present implementations of two key quantum circuits: the three-qubit Toffoli gate and the general two-qubit controlled-unitary gate. Although our experiment is carried out in a photonic architecture, the technique is independent of the particular physical encoding of quantum information, and has the potential for wider application.