Developing quantum computers for real-world applications requires understanding theoretical sources of quantum advantage and applying those insights to design more powerful machines. Toward that end, we introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis (CCPHASE$(theta)$). We estimate the process fidelity for this scheme via Cycle Benchmarking of $mathcal{F}=87.1pm0.8%$, higher than reference two-qubit gate decompositions. CCPHASE$(theta)$ is anticipated to have broad experimental implications, and we report a blueprint demonstration for solving a class of binary constraint satisfaction problems whose construction is consistent with a path to quantum advantage.
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