We introduce a complete transformation rule set and a minimal equation set
for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum
circuits that compute the same Boolean function are reduced to the same normal
form. We can thus easily check the equivalence of circuits by comparing their
normal forms. By applying the Knuth-Bendix completion algorithm to a set of
modified 18 equations introduced by Iwama et al. 2002, we obtain a complete
transformation rule set (i.e., a set of transformation rules with the
properties of `termination' and `confluence'). Our transformation rule set
consists of 114 rules. Moreover, we discovered a minimal combination of
equations for the initial equation set