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Applied Optics

Quaternary Galois field adder based all-optical multivalued logic circuits

Tanay Chattopadhyay, Chinmoy Taraphdar, Jitendra Nath Roy

Published Jul 9, 2009• 37 citations
Publisher

Abstract

Galois field (GF) algebraic expressions have been found to be promising choices for reversible and quantum implementation of multivalued logic. For the first time to our knowledge, we developed GF(4) adder multivalued (four valued) logic circuits in an all-optical domain. The principle and possibilities of an all-optical GF(4) adder circuit are described. The theoretical model is presented and verified through numerical simulation. The quaternary inverter, successor, clockwise cycle, and counterclockwise cycle gates are proposed with the help of the all-optical GF(4) adder circuit. In this scheme different quaternary logical states are represented by different polarized light. A terahertz optical asymmetric demultiplexer interferometric switch plays an important role in this scheme.

DOI
10.1364/ao.48.000e35

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Galois field (GF) algebraic expressions have been found to be promising choices for reversible and quantum implementation of multivalued logic. For the first time to our knowledge, we developed GF(4) adder multivalued (four valued) logic circuits in an all-optical domain. The principle and possibilities of an all-optical GF(4) adder circuit are described. The theoretical model is presented and verified through numerical simulation. The quaternary inverter, successor, clockwise cycle, and counterclockwise cycle gates are proposed with the help of the all-optical GF(4) adder circuit. In this scheme different quaternary logical states are represented by different polarized light. A terahertz optical asymmetric demultiplexer interferometric switch plays an important role in this scheme.
Galois field sum of products (GFSOP) has been found to be very promising for reversible/quantum implementation of multiplevalued logic. In this paper, we show nine quaternary Galois field expansions, using which quaternary Galois field decision diagrams (QGFDD) can be constructed. Flattening of the QGFDD generates quaternary GFSOP (QGFSOP). These QGFSOP can be implemented as cascade of quaternary 1-qudit gates and multi-qudit Feynman and Toffoli gates. We also show the realization of quaternary Feynman and Toffoli gates using liquid ion-trap realizable 1-qudit gates and 2-qudit Muthukrishnan-Stroud gates. Besides the quaternary functions, this approach can also be used for synthesis of encoded binary functions by grouping 2-bits together into quaternary value. For this purpose, we show binary-to-quaternary encoder and quaternary-to-binary decoder circuits using quaternary 1-qudit gates and 2-qudit Muthukrishnan-Stroud gates.
Galois field sum of products (GFSOP) has been found to be very promising for reversible/quantum implementation of multiple-valued logic. In this paper, we show ten quaternary Galois field expansions, using which quaternary Galois field decision diagrams (QGFDD) can be constructed. Flattening of the QGFDD generates quaternary GFSOP (QGFSOP). These QGFSOP can be implemented as cascade of quaternary 1-qudit gates and multi-qudit Feynman and Toffoli gates. We also show the realization of quaternary Feynman and Toffoli gates using liquid ion-trap realizable 1-qudit gates and 2-qudit Muthukrishnan-Stroud gates. Besides the quaternary functions, this approach can also be used for synthesis of encoded binary functions by grouping 2-bits together into quaternary value. For this purpose, we show binary-to-quaternary encoder and quaternary-to- binary decoder circuits using quaternary 1-quidit gates and 2-qudit Muthukrishnan-Stroud gates.
In this paper, all-optical scheme for designing some polarization encoded quaternary logic gates (quaternary min and quaternary delta literal) with the help of Mach Zender interferometric (MZI) switch have been discussed. With the demanding scenario of communication and optical computing technology the quadruple valued logic systems are the most important ones in the multi valued logic system. In order to overcome the electronic bottlenecks and fully exploit the advantages of optics, multi valued logic (MVL) is explore here for data communication system.
Recently design of reversible logic circuits draw much attention in nanotechnology and quantum computing. Conventional logic circuits are not reversible. A reversible circuit maps each input vector, into a unique output vector and vice versa. In this paper, we propose a novel scheme of Toffoli and Feynman gates in all-optical domain using terahertz optical asymmetric demultiplexer (TOAD). Both the realizations are supported through proper flowchart followed by simulation in Macad-7. Simulation results verify the functionality of both the gates along with verified reversibility. Toffoli and Feynman gates can be used to implement Boolean expressions and arithmetic operations.
Quantum technology is one of the most promising technologies for future computing systems, since quantum algorithms solve problems much more efficiently than classical algorithms. All quantum algorithms are made up of quantum logic circuits. A quantum logic circuit is made up of quantum gates (reversible in nature) and is designed using reversible logic synthesis methods. For a given Hilbert space, ternary quantum system requires 0.63 times qutrits than the corresponding number of qubits. Thus, the ternary quantum system provides a much more compact and efficient information encoding. Beside other technologies, ternary quantum logic system can be realized using photon polarization. These advantages of ternary quantum system open avenue for developing ternary quantum algorithms. Galois field sum of products (GFSOP) based synthesis of ternary quantum logic circuit is the most practical approach, since any ternary logic function with many inputs can be represented as GFSOP expression and the GFSOP expression can be implemented as cascade of ternary quantum gates. Here we discuss minimization of ternary logic function as GFSOP expression using quantuminspired evolutionary algorithm. We also discuss a method of realization of ternary GFSOP expression using ternary quantum gates. Experimental results are given to show the effectiveness of the ternary GFSOP minimization technique.
In recent years reversible logic has emerged as a promising computing model for applications in dissipation less optical computing, low power CMOS, quantum computing, etc. In reversible circuits there exist a one-to-one mapping between the inputs and the outputs resulting in no loss of information. Researchers have implemented reversible logic gates in optical computing domain as it can provide high speed and low energy requirement along with easy fabrication at the chip level [1]. The all optical implementation of reversible gates are based on semiconductor optical amplifier (SOA) based Mach-Zehnder interferometer (MZI) due to its significant advantages such as high speed, low power, fast switching time and ease in fabrication. In this work we present the all optical implementation of an n bit reversible ripple carry adder for the first time in literature. The all optical reversible adder design is based on two new optical reversible gates referred as optical reversible gate I (ORG-I) and optical reversible gate II (ORG-II) and the existing all optical Feynman gate. The two new reversible gates ORG-I and ORGI-I are proposed as they can implement a reversible adder with reduced optical cost which is the measure of number of MZIs switches and the propagation delay, and with zero overhead in terms of number of ancilla inputs and the garbage outputs. The proposed all optical reversible adder design based on the ORG-I and ORG-II reversible gates are compared and shown to be better than the other existing designs of reversible adder proposed in non-optical domain in terms of number of MZIs, delay, number of ancilla inputs and the garbage outputs. The proposed all optical reversible ripple carry adder will be a key component of an all optical reversible ALU that can be applied in a wide variety of optical signal processing applications.