  We present "Diagrams of States", a way to graphically represent and analyze
how quantum information is elaborated during the execution of quantum circuits.
This introductory tutorial illustrates the basics, providing useful examples of
quantum computations: elementary operations in single-qubit, two-qubit and
three-qubit systems, immersions of gates on higher dimensional spaces,
generation of single and multi-qubit states, procedures to synthesize unitary,
controlled and diagonal matrices. To perform the analysis of quantum processes,
we directly derive diagrams of states from physical implementations of quantum
circuits associated to the processes. Complete diagrams are then rearranged
into simplified diagrams, to visualize the overall effects of computations.
Conversely, diagrams of states help to conceive new quantum algorithms, by
schematically describing desired manipulations of quantum information with
intuitive diagrams and then by guessing the equivalent complete diagrams, from
which the corresponding quantum circuit is obtained effortlessly. Related
examples and analysis of complex algorithms will be provided in future works,
for whose comprehension this first tutorial offers the necessary introduction.
