  Quantum computation architecture based on $d$-level systems, or qudits, has
attracted considerable attention recently due to their enlarged Hilbert space.
Extensive theoretical and experimental studies have addressed aspects of
algorithms and benchmarking techniques for qudit-based quantum computation and
quantum information processing. Here, we report a physical realization of qudit
with upto 4 embedded levels in a superconducting transmon, demonstrating
high-fidelity initialization, manipulation, and simultaneous multi-level
readout. In addition to constructing SU($d$) operations and benchmarking
protocols for quantum state tomography, quantum process tomography, and
randomized benchmarking etc, we experimentally carry out these operations for
$d=3$ and $d=4$. Moreover, we perform prototypical quantum algorithms and
observe outcomes consistent with expectations. Our work will hopefully
stimulate further research interest in developing manipulation protocols and
efficient applications for quantum processors with qudits.
