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IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Multi-valued Quantum Cascade Realization with Group Decomposition

Miroslav Saraivanov, Marek A. Perkowski

Published May 1, 2018• 0 citations
Publisher

Abstract

This paper introduces a method for realizing binary controlled inputs to p-valued single output functions (where p is prime, multi-valued logic state, in particular p=2, 3, or 5) by using a canonical (reversible, quantum) cascade of quantum logic blocks derived from decomposed group functions and the Walsh spectrum. The first part of our method is based on a publication by T. Sasao in [4] and is then extended to classical reversible logic and quantum cascades optimized for ion trap and quantum dot technologies.

DOI
10.1109/ismvl.2018.00050

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This paper introduces a method for realizing binary controlled inputs to p-valued single output functions (where p is prime, multi-valued logic state, in particular p=2, 3, or 5) by using a canonical (reversible, quantum) cascade of quantum logic blocks derived from decomposed group functions and the Walsh spectrum. The first part of our method is based on a publication by T. Sasao in [4] and is then extended to classical reversible logic and quantum cascades optimized for ion trap and quantum dot technologies.
This paper extends the decomposition from the group theory based methods of Sasao and Saraivanov to design binary input multivalued output quantum cascades realized with optical NOT, SWAP, and Fredkin Gates. We present this method for 3, 5, and 7 valued outputs, but in general it can be used for odd prime valued outputs. The method can be extended to realize hybrid functions with different valued outputs. A class of local transformations is presented that can simplify the final cascade circuits. Using these simplifying transformations, we present an upper bound on the maximum number of gates in an arbitrary nnn-variable input and kkk-valued output function.
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.
This paper extends the decomposition from the group theory based methods of Sasao and Saraivanov to design binary input multivalued output quantum cascades realized with optical NOT, SWAP, and Fredkin Gates. We present this method for 3, 5, and 7-valued outputs, but in general it can be used for odd prime-valued outputs. The method can be extended to realize hybrid functions with different valued outputs. A class of local transformations is presented that can simplify the final cascade circuits. Using these simplifying transformations, we present an upper bound on the maximum number of gates in an arbitrary nnn-variable input and kkk-valued output function.
The decomposition from the group theory-based methods of Sasao and Saraivanov is extended to design binary quantum cascades, using the quantum rotational gates by the X-axis (CNOT and RX), Y-axis (RY), and Z-axis (controlled-Z) of the Bloch sphere. A class of local transformations is also presented to simplify the final canonical cascade circuits. Our proposed methodology is well suited for quantum layouts, as each single-qubit gate has one target qubit and each double-qubit gate has one control qubit and one target qubit, thereby never creating a graph of triangular connectivity.
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).
We consider the symmetric group S-n in the special case where n = pq (both p and q being integer). Applying Birkhoff's theorem, we prove that an arbitrary element of S-pq can be decomposed into a product of three permutations, the first and the third being elements of the Young subgroup S-p(q), whereas the second one is an element of the dual Young subgroup S-p(q). This leads to synthesis methods for arbitrary multiple-valued reversible logic circuits of logic width w. These circuits indeed form a group isomorphic to Sr-w, where r is the radix of the multiple-valued logic. A particularly efficient decomposition is found by choosing p = r and thus q = r(w-1). As a result, an arbitrary reversible logic circuit of radix r and width w is decomposed into a cascade of 2w - 1 control gates, i.e. logic building blocks, which manipulate only one of the w dits.