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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

GF(4) Based Synthesis of Quaternary Reversible/Quantum Logic Circuits

Mozammel H. A. Khan, Marek A. Perkowski

Published May 1, 2007• 30 citations
Publisher

Abstract

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.

DOI
10.1109/ismvl.2007.26

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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) 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.
Multiple-valued logic functions having many input variables can be easily expressed as Galois field sum of products (GFSOP) expression and can be realized using cascade of multiple-valued Feynman and Toffoli gates (Khan et al., 2005). Conventional binary functions can be expressed very easily as quaternary functions by grouping 2-bits together. These quaternary functions can be expressed as quaternary Galois field sum of products expression and can be implemented as cascade of quaternary Feynman and Toffoli gates. These gates are macro-level gates and need to be realized using technology based primitive gates. In this paper, we show the realization of quaternary Feynman and Toffoli gates on the top of theoretically liquid ion-trap realizable 1-qudit and 2-qudit Muthukrishnan-Stroud gates (Muthukrishnan, 2000).
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.
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.
Even though there are 256 possible 1-qudit(1-variable quantum digit) functions in quaternary logic, the most useful functions are 4!=24 ones capable of representing in QGFSOP expressions by possible permuting of 0,1,2, and 3. In this paper, we propose a permutational literal(PL) representation and a QPL(Quaternary PL) gate which use the operands of a multiplicand A and an augend D in +D(GF4) operation as a control variable of multi-cascaded PLs. And we also present new PL synthesis algorithms to synthesize QGFSOP expressed 24 (1-qudit) functions by applying three PL operators as ab(mutual permutation), + D(addition), and XA (multiplication). Finally architectures, circuits, and a CMOS implementation to realize proposed PL synthesis algorithms for +D(GF4) functions are presented.
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.