Transformation rules for designing CNOT-based quantum circuits

This paper gives a simple but nontrivial set of local transformation rules for control-NOT (CNOT)-based combinatorial circuits. It is shown that this rule set is complete, namely, for any two equivalent circuits, S₁ and S₂, there is a sequence of transformations, each of them in the rule set, which changes S₁ to S₂. Our motivation is to use this rule set for developing a design theory for quantum circuits whose Boolean logic parts should be implemented by CNOT-based circuits. As a preliminary example, we give a design procedure based on the transformation rules which reduces the cost of CNOT-based circuits.