Quantum computation is conventionally performed using binary quantum logic by operating on quantum bits, or qubits. However, most quantum devices naturally have multiple accessible energy levels beyond the lowest two traditionally used to define a qubit. While unintentional occupation of these higher energy states introduces computational errors, intentional use of these states can provide benefits that are potentially useful in the noisy quantum compute era. We present how, using Qiskit Pulse, one can program a quantum computer over the cloud at the pulse level and implement ternary logic operations on fixed-frequency transmons. From this, we realize a three-qubit Toffoli gate using two different decompositions with 50% fewer (3 instead of 6) two-transmon gates, when compared with the usual binary implementation using CNOT gates.We discuss the theoretical requirements for such qutrit decompositions and our experimental progress of implementing them on pulse-control-enabled devices made available by IBM Quantum. We highlight the advantages and disadvantages of using higher-energy states to obtain optimized circuits, compared to default qubit realizations. Our work encourages an investigation of cloud-based quantum circuit optimization techniques using multi-level quantum systems.