  This paper deals with the size complexity of minimal {\it two-way quantum
finite automata} (2qfa's) necessary for operations to perform on all inputs of
each fixed length. Such a complexity measure, known as state complexity of
operations, is useful in measuring how much information is necessary to convert
languages. We focus on intersection, union, reversals, and catenation
operations and show some upper bounds of state complexity of operations on
2qfa's. Also, we present a number of non-regular languages and prove that these
languages can be accepted by 2qfa's with one-sided error probabilities within
linear time. Notably, these examples show that our bounds obtained for these
operations are not tight, and therefore worth improving. We give an instance to
show that the upper bound of the state number for the simulation of one-way
deterministic finite automata by two-way reversible finite automata is not
tight in general.
