  We analyze the depth of the memory of quantum memory channels generated by a
fixed unitary transformation describing the interaction between the principal
system and internal degrees of freedom of the process device. We investigate
the simplest case of a qubit memory channel with a two-level memory system. In
particular, we explicitly characterize all interactions for which the memory
depth is finite. We show that the memory effects are either infinite, or they
disappear after at most two uses of the channel. Memory channels of finite
depth can be to some extent controlled and manipulated by so-called reset
sequences. We show that actions separated by the sequences of inputs of the
length of the memory depth are independent and constitute memoryless channels.
