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Engineering letters

Design of Reversible/Quantum Ternary Comparator Circuits

Mozammel H. A. Khan

Published Jan 1, 2008• 19 citations
OpenAlex

Abstract

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.

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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.
Abstract Multi-valued quantum systems have more significant advantages in computational performance than binary quantum system, since the circuit design is more compact and efficient in multi-valued quantum systems. Hence, synthesizing multi-valued logic circuits is a better choice for future quantum computing technology. A subcircuit in the Grover’s algorithm is called an oracle, which inputs a set of quantum qudits and gives an output according to the search conditions. Greater-than/less-than/equality comparators are widely used in search conditions. In this paper, the ternary Toffoli gate proposed by Monfared and Haghparast is extended to multi-valued under certain conditions, and the existing optimal quaternary n-qudit comparator is optimized using the multi-valued Toffoli gate. In addition, we design universal multi-valued equality/greater-than/less-than comparators and complementary component, which constitute a comprehensive multi-valued n-qudit comparator. Compared to existing comparator circuits, these comparator circuits are more efficient in terms of quantum cost, garbage output, hardware complexity, and constant input.
Circuit design with no energy dissipation is possible in the reversible logic. Also, using the quantum multiple‐valued logic in the design of circuits due to the reduction of circuit width leads to the smaller chip area and lower power consumption. Quaternary reversible logic potentially has several advantages in comparison to the equivalent ternary and binary quantum technologies. Hence, the quaternary quantum technology is a promising choice for the future digital system design. Here, the authors proposed an efficient design of 1‐qudit and n ‐qudit quantum comparators. The comparator module is widely used in search conditions, such as Grover's quantum search algorithm in quantum computing. The proposed circuits were better than their existing counterparts in terms of quantum cost, the number of constant inputs, the number of garbage outputs, and hardware complexity. To design these circuits, they have used 1‐qudit gates and 2‐qudit Muthukrishnan–Stroud gates. The efficiency of their proposed circuits was reported.
Methods of quantum/reversible logic synthesis are based on the use of the binary nature of quantum computing. However, multiple-valued logic is a promising choice for future quantum computer technology due to a number of advantages over binary circuits. In this paper we have developed a synthesis of ternary reversible circuits based on Muthukrishnan-Stroud gates using a genetic algorithm. The method of coding chromosome is presented, and well-grounded choice of algorithm parameters allowed obtaining better circuit schemes of one- and n-qutrit ternary comparators compared with other methods. These parameters are quantum cost of received reversible devices, delay time and number of constant input (ancilla) lines. Proposed implementation of the genetic algorithm has led to reducing of the device delay time and the number of ancilla qutrits to 1 and 2n-1 for one- and n-qutrits full comparators, respectively. For designing of n-qutrit comparator we have introduced a complementary device which compares output functions of 1-qutrit comparators.
Multiple-valued logic is a promising choice for future computer technologies, which provides a set of advantages comparing to binary circuits. We have developed an adaptive genetic algorithm for ternary reversible circuits using Muthukrishnan-Stroud gates. The method for chromosomes coding, as well as a reasonable choice of algorithm parameters, allowed obtaining circuits for ternary arithmetic devices, which are better than other known methods in terms of quantum cost, delay time and amount of input ancillary and output garbage qutrits. Based on our realization of ternary parallel full-adder we synthesized reversible ternary parallel adder/subtractor, which has better parameters over the previously reported devices.
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.
Typical methods of quantum/reversible synthesis are based on using the binary character of quantum computing.However, multi-valued logic is a promising choice for future computer technologies, given a set of advantages when comparing to binary circuits.In this work, we have developed a genetic algorithm-based synthesis of ternary reversible circuits using Muthukrishnan-Stroud gates.The method for chromosomes coding that we present, as well as a judicious choice of algorithm parameters, allowed obtaining circuits for half-adder and full adder which are better than other published methods in terms of cost, delay times and amount of input ancillary bits.A structure of the circuits is analyzed in details, based on their decomposition.