Most quantum computing theory developed thus far has focused on qubits - two-level quantum systems. However, there has been a recent surge of interest in studying the more general case of $d$-level quantum systems, called qudits. This has led to applications of qudits for quantum algorithms, improving magic state distillation noise thresholds, and communication noise resilience, as well as experimental realizations of qudits. In a series of recent papers, we explore qutrit ($d = 3)$ compilation, not only extending several known qubit gate constructions to qutrits, but also discovering several curious cases of where qubits and qutrits differ.

First, we find an explicit construction for any qutrit multiple-controlled Toffoli. Moreover, we show how for qutrits, given any Clifford+$T$ unitary, we can add any number of control qutrits unitarily and without ancillae in Clifford+$T$; in the qubit setting, this result does not hold. All these constructions are ancilla-free and have polynomial gate count in the qutrit Clifford+$T$ gate set. We further present results on exact synthesis of controlled phase gates and arbitrary diagonal unitaries in a Clifford+single-qutrit phase gates gate set. Our recent results have elevated our understanding of qutrit circuits from a mixed bag of neat tricks to full-fledged results applicable to approximately universal gate sets.