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In this paper we present a method for minimizing reversible quantum circuitsusing the Quantum Operator Form (QOF); a new representation of quantum circuitand of quantum-realized reversible circuits based on the CNOT, CV andCV$^\dagger$ quantum gates. The proposed form is a quantum extension to thewell known Reed-Muller but unlike the Reed-Muller form, the QOF allows theusage of different quantum gates. Therefore QOF permits minimization of quantumcircuits by using properties of different gates than only the multi-controlToffoli gates. We introduce a set of minimization rules and a pseudo-algorithmthat can be used to design circuits with the CNOT, CV and CV$^\dagger$ quantumgates. We show how the QOF can be used to minimize reversible quantum circuitsand how the rules allow to obtain exact realizations using the above mentionedquantum gates.

Minimization of Quantum Circuits using Quantum Operator Forms