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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Synthesis of GF(3) Based Reversible/Quantum Logic Circuits without Garbage Output

Mahmud Muntakim Khan, Ayan Kumar Biswas, Shuvro Chowdhury, Masud Hasan, Asif Islam Khan

Published May 1, 2009• 15 citations
Publisher

Abstract

We present a method of synthesizing ternary Galois field (GF(3)) based reversible/quantum logic circuits without any ancillary trits/qutrits and hence without any garbage outputs. We realize multi input ternary Toffoli gate and square functions of GF(3) variables using linear ion trap realizable Muthukrishnan-Stroud (M-S) gates and shift gates in the absence of ancillary qutrits. Then based on the Galois Field Sum of Products (GFSOP)expression of a multi-variable GF(3) function, we synthesize the corresponding circuit.

DOI
10.1109/ismvl.2009.73

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We present a method of synthesizing ternary Galois field (GF(3)) based reversible/quantum logic circuits without any ancillary trits/qutrits and hence without any garbage outputs. We realize multi input ternary Toffoli gate and square functions of GF(3) variables using linear ion trap realizable Muthukrishnan-Stroud (M-S) gates and shift gates in the absence of ancillary qutrits. Then based on the Galois Field Sum of Products (GFSOP)expression of a multi-variable GF(3) function, we synthesize the corresponding circuit.
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.
Galois Field Sum of Products (GFSOP) leads to efficient multi-valued reversible circuit synthesis using quantum gates. In this paper, we propose a new generalization of ternary Toffoli gate and another new generalized reversible ternary gale with discussion of their quantum realizations. Algorithms for synthesizing ternary GFSOP using quantum cascades of these gates are proposed In both the synthesis methods, 5 ternary shift operators and ternary swap gate are used We also propose quantum realizations of 5 ternary shift operators and ternary swap gate. In the cascades of the new ternary gates, local mirrors, variable ordering, and product ordering techniques are used to reduce the circuit cost. Experimental results show that the cascade of the new ternary gates is more efficient than the cascade of ternary Toffoli gates.
Reversible quantum computer system is one of the best choices for future computer systems. Multiple-valued logic especially ternary logic is a good candidate for the realization of reversible quantum computer. An efficient logic synthesis mechanism is essential for the low-cost realization. Toffoli gate is an important gate for quantum logic synthesis. It is the basic element for the Galois Field Sum of Product (GFSOP) expression based logic synthesis mechanism. So, for low-cost realization of any ternary logic function, a low-cost implementation of Toffoli gate is very necessary. This paper shows a low-cost, practically realizable, and efficient realization of 3-qutrit ternary Toffoli gate by using ion-trap realizable Muthukrishnan-Stroud gate. This realization is more efficient and less costly than other realizations.
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.
Reversible/quantum multiple-valued logic circuit has several advantages over reversible/quantum binary logic circuit. Galois field sum of products (GFSOP) based synthesis of multiple-valued logic function is more promising and practical than other approaches. In this paper, we have developed 196 Galois field expansions (GFE) and have proposed a method of minimization of GFSOP expression for non-reversible quinary logic function using the application of these GFEs. We have also proposed a method of realization of quinary GFSOP expression as cascade of quinary reversible/quantum gates. Experimental results with 26 functions having up to six inputs and two outputs show that a significant minimization can be achieved using the proposed minimization method.