Contemporary quantum computers encode and process quantum information in binary qubits $(d=2)$. However, many architectures include higher energy levels that are left as unused computational resources. We demonstrate a superconducting ququart $(d=4)$ processor and combine quantum optimal control with efficient gate decompositions to implement high-fidelity ququart gates. We distinguish between viewing the ququart as a generalized four-level qubit and an encoded pair of qubits, and characterize the resulting gates in each case. In randomized benchmarking experiments we observe gate fidelities $\ge 95\%$ and identify coherence as the primary limiting factor. Our results validate ququarts as a viable tool for quantum information processing.

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• Quantum benchmarking
• Quantum control
• Quantum gates
• Quantum information processing
• Quantum information with solid state qubits
• Quantum tomography
• Qudits
• Superconducting qubits