  Gate-based quantum computers typically encode and process information in
two-dimensional units called qubits. Using $d$-dimensional qudits instead may
offer intrinsic advantages, including more efficient circuit synthesis,
problem-tailored encodings and embedded error correction. In this work, we
design a superconducting qudit-based quantum processor wherein the logical
space of transmon qubits is extended to higher-excited levels. We propose a
universal gate set featuring a two-qudit cross-resonance entangling gate, for
which we predict fidelities beyond $99\%$ in the $d=4$ case of ququarts with
realistic experimental parameters. Furthermore, we present a decomposition
routine that compiles general qudit unitaries into these elementary gates,
requiring fewer entangling gates than qubit alternatives. As proof-of-concept
applications, we numerically demonstrate the synthesis of ${\rm SU}(16)$ gates
for noisy quantum hardware and an embedded error correction sequence that
encodes a qubit memory in a transmon ququart to protect against pure dephasing
noise. We conclude that universal qudit control -- a valuable extension to the
operational toolbox of superconducting quantum information processing -- is
within reach of current transmon-based architectures and has applications to
near-term and long-term hardware.
