Low quantum cost realization of generalized peres and toffoli gates with multiple-control signals

Multiple-control Toffoli gates are basic building blocks for reversible and quantum circuits. 3-bit Peres gate can be considered as a pair of 3-bit and 2-bit Toffoli gates, which can be implemented with fewer elementary quantum gates (what defines a metric called quantum cost) than the total number of elementary quantum gates required by each of Toffoli gates implemented separately. Due to this property, Peres gates are often used for quantum cost reduction of reversible circuits. This paper introduces a generalization of the Peres gate with n > 2 control signals. For such generalized gates corresponding quantum circuits are presented. The realizations have quantum cost of n² and require no ancillary signals. The proposed designs are built from elementary quantum gates, which consist of NOT, controlled NOT, controlled `square-roots-of-NOT' (known as controlled V/V<sup>+</sup> gates) and controlled `higher roots-of-NOT' (up to 2^n-1-roots-of-NOT). The obtained quantum circuits are not entangled. The proposed designs are used to construct Toffoli gates of n control signals with quantum cost of 2n² - 2n + 1 and using no ancillary signals. This is an improvement in comparison with 2ⁿ⁺¹ - 3 and 48n² + O(n), which are quantum costs of the known designs of the multiple-control Toffoli gate without ancillary signals.