This paper introduces a logic structure for reversible quantum circuit synthesis. Our synthesis method aims to minimize the quantum cost of reversible quantum circuits with decoders. In this method, multivalued input, binary output (MVI) functions are utilized as a mathematical concept only, but the circuits are binary. We introduce the concept of “MultiValued Input Fixed Polarity Reed-Muller” forms. Our decoder-based circuit uses three logical levels in contrast with commonly used methods based on Exclusive-or Sum of Products (ESOP) with two levels (AND-XOR expressions), realized by Toffoli gates. In general, the high number of input qubits in the resulting Toffoli gates is a problem that greatly impacts the quantum cost. Using decoders decreases the number of input qubits in these Toffoli gates. We present two practical algorithms for three-level circuit synthesis by finding the MVI–Fixed-Polarity Reed-Muller (MVI-FPRM): product-matching and the newly developed butterfly diagrams. The best MVI-FPRM forms are factorized and reduced to approximate MultiValued Input Generalized Reed-Muller (MVI-GRM) forms. Analytically, various multi-output binary functions, such as comparators, are tested using both the proposed methods and the ESOP approach, and they are evaluated using two cost metrics. The evaluation shows that our method generally has a smaller cost value than that of the ESOP-based approach. Notably, the results demonstrate that our method becomes increasingly effective with comparators of greater sizes.

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• Quantum circuits
• Quantum gates
• Quantum information processing with continuous variables
• Qudits
• Synthesis