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arXiv:2010.00085

Realization of p_valued Deutsch quantum gates

Claudio Moraga

Published Sep 30, 2020• 0 citations
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Abstract

In this report reversible Toffoli and quantum Deutsch gates are extended to the p_valued domain. Their structural parameters are determined and their behavior is proven. Both conjunctive and disjunctive control strategies with positive and mixed polarities are introduced for the first time in a p_valued domain. The design is based on elementary Muthukrishnan_Stroud quantum gates, hence the realizability of the extended gates in the context of ion traps should be possible.

DOI
10.48550/arxiv.2010.00085

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In this report reversible Toffoli and quantum Deutsch gates are extended to the p_valued domain. Their structural parameters are determined and their behavior is proven. Both conjunctive and disjunctive control strategies with positive and mixed polarities are introduced for the first time in a p_valued domain. The design is based on elementary Muthukrishnan_Stroud quantum gates, hence the realizability of the extended gates in the context of ion traps should be possible.
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.
This paper presents basic aspects of non-binary reversible and quantum computing in a p-valued environment, since there are no physical reasons for quantum computing to be necessarily binary. A “quantum technology” is not yet available, but different alternatives at the level of laboratory experiments are promising. A theoretical background is needed to face the challenge of designing circuits for quantum computing. Pauli matrices are introduced in the p-valued domain and their properties are explained. The Vilenkin-Chrestenson matrix is shown to produce superposition of states. Entanglement of quantum states in the p-valued environment is introduced and its measurement effect is illustrated. It is shown that in the p-valued domain, the binary Toffoli gate may be given different generalizations with different functionalities. An ancillary-free realization of a new generalized p-valued Toffoli gate based on Muthukrishnan-Stroud elementary gates is presented, its functionality is extended to two kinds of control, and a proof of its performance is given.
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.