  Compared with a qubit, a qudit (i.e., $d$-level or $d$-state quantum system)
provides a larger Hilbert space to store and process information. On the other
hand, qudit-based hybrid quantum computing usually requires performing hybrid
quantum gates with qudits different in their nature or in their encoding
format. In this work, we consider the qutrit case, i.e., the case for a qudit
with $d$=3. We propose a simple method to realize a hybrid quantum
controlled-SUM gate with one superconducting (SC) qutrit and a cat-state
qutrit. This gate plus single-qutrit gates form a universal set of ternary
logic gates for quantum computing with qutrits. Our proposal is based on
circuit QED and operates essentially by employing a SC ququart (a four-level
quantum system) dispersively coupled to a microwave cavity. The gate
implementation is quite simple because it only requires a single basic
operation. Neither classical pulse nor measurement is needed. The auxiliary
higher energy level of the SC ququart is virtually excited during the gate
operation, thus decoherence from this level is greatly suppressed. As an
application of this gate, we discuss the generation of a hybrid
maximally-entangled state of one SC qutrit and one cat-state qutrit. We further
analyze the experimental feasibility of creating such hybrid entangled state in
circuit QED. This proposal is quite general and can be extended to accomplish
the same task in a wide range of physical system, such as a four-level natural
or artificial atom coupled to an optical or microwave cavity.
