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arXiv:1503.07356

Mixed polarity reversible Peres gates

Claudio Moraga

Published Mar 25, 2015• 0 citations
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Abstract

Reversible Peres gates with more than two all over binary-valued control signals are discussed. Methods are disclosed for the low cost realization of this kind of Peres gates without requiring ancillary lines. Proper distribution of the controlled gates and their inverses allow driving the reversible Peres gate with control signals of different polarities.

DOI
10.48550/arxiv.1503.07356

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•Jan 1, 2012•0 citations
Reversible Peres gates with more than two all over binary-valued control signals are discussed. Methods are disclosed for the low cost realization of this kind of Peres gates without requiring ancillary lines. Proper distribution of the controlled gates and their inverses allow driving the reversible Peres gate with control signals of different polarities.
Reversible Peres gates with more than two all over binary‐valued control signals are discussed. Methods are disclosed for the low‐cost realisation of this kind of Peres gates without requiring ancillary lines. It is shown that Peres gates with n control signals may be obtained with a quantum cost of 2 n + 1 − n − 2, using Feynman gates and controlled gates realising the κ th root of NOT, where κ = 2 n −1 . Proper distribution of the controlled gates and their inverses allows driving the reversible Peres gate with control signals of different polarities.
The paper introduces dual Toffoli and Peres reversible gates, which operate under disjunctive control, and shows their functionality based on the Barenco et al. quantum model. Both uniform and mixed polarity are considered for the controls. Rewriting rules are presented, which provide a possible reduction of the number of gates and quantum cost of reversible (sub)circuits using standard Toffoli or Peres gates. Finally, a Clifford+T realization of a dual Toffoli and a dual Peres gate is shown, which may be used when mapping reversible circuits to the IBM quantum computers.
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 n2and 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+gates) and controlled `higher roots-of-NOT' (up to 2n-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.
Optimal synthesis of reversible circuits is a very hard task. In 2010, a method and a tool of practical usage for finding optimal circuits built from multi-control Toffoli gates for any 4-bit reversible specification were finally developed. In 2011 we published an extension to the algorithm, which allows to reduce the quantum cost of the resulting circuits. In this paper we present another extension to this approach. Namely, we have extended the reversible gate library to mixed-polarity multi-control Toffoli gates (i.e. with both positive and negative controls). Our experimental results for the known reversible benchmarks show that using mixed-polarity Toffoli gates gives significant savings in gate count. The paper presents standard computational experiments for circuit synthesis. These results have implications in testing synthesis algorithms for reversible mixed-polarity circuits and quantum circuits.
Optimal synthesis of reversible circuits is a very hard task. In 2010, a method and a tool of practical usage for finding optimal circuits built from multi-control Toffoli gates for any 4-bit reversible specification were finally developed. In 2011 we published an extension to the algorithm, which allows to reduce the quantum cost of the resulting circuits. In this paper we present another extension to this approach. Namely, we have extended the reversible gate library to mixed-polarity multi-control Toffoli gates (Le. with both positive and negative controls). Our experimental results for the known reversible benchmarks show that using mixed-polarity Toffoli gates gives significant savings in gate count. The paper presents standard computational experiments for circuit synthesis. These results have implications in testing synthesis algorithms for reversible mixed-polarity circuits and quantum circuits.