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International Conference on Computer and Information Technology (ICCIT)

Low-cost realization of toffoli gate for the low-cost synthesis of quantum ternary logic functions

Md. Mehedi Hasan

Published Dec 1, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.1109/iccitechn.2008.4803024

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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.
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.
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).
Multiple-valued Feynman and Toffoli gates are used in GFSOP based synthesis of quantum logic circuits. These gates are macro-level gates and need to be realized using technology dependent primitive gates. In this paper, we present ancilla input free architectures for realization of d-valued (dges3 ) Feynman and n-qudit (nges3 ) Toffoli gates on the top of liquid ion-trap realizable Muthukrishnan-Stroud gates. The proposed architectures can be used for any d if GF(d) can be constructed. We show realization examples of ternary, quaternary, and quinary Feynman gates, and 3 and 4-qudit Toffoli gates. The present realizations require either less or equal primitive gates than the previously reported realizations. Moreover, in contrast to the earlier realizations, the present Toffoli gate realizations do not require any ancilla input, which reduce the register width of a synthesized quantum logic circuit.
We present realization of ternary Toffoli gate and modified Fredkin gate for quantum computing on top of ion trap realizable Muthukrishnan-Stroud primitive gates. Our design methodology is based on first realizing the quantum circuits using generalized ternary gates and Feynman gates and then replacing them with their equivalent realization using Muthukrishnan-Stroud gates. Our realization of ternary Toffoli gate is more efficient than the previously published result and modified Fredkin gate is realized for the first time in literature using 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.