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International Conference on Computer and Information Technology (ICCIT)

Synthesis of incompletely specified multi-output quaternary function using quaternary quantum gates

Mozammel H. A. Khan

Published Dec 1, 2007• 2 citations
Publisher

Abstract

Multiple-valued quantum circuits are a promising choice for future quantum computing technology since they have several advantages over binary quantum circuits. In this paper, we propose quaternary multi-qudit controlled gate family and show their realization on the top of liquid ion-trap realizable 1-qudit gates and 2-qudit Muthukrishnan-Stroud gates. Then we show a method of synthesizing incompletely specified multi-output quaternary function using quaternary 1-qudit gates and multi-qudit controlled gates.

DOI
10.1109/iccitechn.2007.4579405

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Multiple-valued quantum circuits are a promising choice for future quantum computing technology since they have several advantages over binary quantum circuits. In this paper, we propose quaternary multi-qudit controlled gate family and show their realization on the top of liquid ion-trap realizable 1-qudit gates and 2-qudit Muthukrishnan-Stroud gates. Then we show a method of synthesizing incompletely specified multi-output quaternary function using quaternary 1-qudit gates and multi-qudit controlled 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).
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.
We present a new type of quantum realizable reversible cascade. Next we present a new algorithm to synthesize arbitrary single-output ternary functions using these reversible cascades. The cascades use “Generalized Multi-Valued Gates” introduced here, which extend the concept of Generalized Ternary Gates introduced previously. While there were 216 GTGs, a total of 12 ternary gates of the new type are sufficient to realize arbitrary ternary functions. (The count can be further reduced to 5 gates, three 2-qubit and two 1-qubit). Such gates are realizable in quantum ion trap devices. For some functions, the algorithm requires fewer gates than results previously published [1, 5, 8, 14]. In addition, the algorithm also does conversion from arbitrary ternary logic to reversible logic at the cost of relatively small garbage. The algorithm is implemented here in ternary logic, but generalization to arbitrary radix is both straightforward and sees a reduction in growth of cost as the radix is increased.
Abstract—Multiple-valued quantum circuits are promising choices for future quantum computing technology, since the multiple-valued quantum system is more compact than the corresponding binary quantum system. Grover’s quantum search algorithm requires a sub-circuit called oracle, which takes a set of inputs and gives an output stating whether a given search condition on the inputs is satisfied or not. Equality, less-than, and greater-than comparisons are widely used as search conditions. In this paper, we show design of quantum ternary equality, less-than, and greater-than comparators on the top of ion-trap realizable 1-qutrit gates and 2-qutrit Muthukrishnan-Stroud gates. Index Terms—Multiple-valued logic, quantum logic, reversible logic, ternary comparators, ternary logic I.