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International Journal of Digital Content Technology and its Applications

Efficient Synthesis of Optimal 3-qubit Reversible Circuits Using Bit Operation

Zhiqiang Li, Hanwu Chen, Xiaoyu Song B

Published Jun 30, 2012• 1 citation
Publisher
DOI
10.4156/jdcta.vol6.issue11.50

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We use bit operation to construct the novel and perfect Hash function and present an efficient algorithm which can construct optimal quantum reversible logic circuits with various types of gates by using the Hash table and produce quantum circuits with minimal cost in theory. Judging by the internationally recognized reversible functions of three variables, the algorithm not only synthesizes all optimal reversible logic circuits, but also runs extremely faster than other ones. The experimental results show that the average speed of the algorithm which synthesizes circuit with minimum length is 69.8 times that of currently best result [9].
Owing to the exponential nature of the memory and run-time complexity, many methods can only synthesize 3-bit reversible circuits and cannot synthesize 4-bit reversible circuits well. We mainly absorb the ideas of our 3-bit synthesis algorithms based on hash table and present the efficient algorithms which can construct almost all optimal 4-bit reversible logic circuits with many types of gates and at mini-length cost based on constructing the shortest coding and the specific topological compression; thus, the lossless compression ratio of the space of n -bit circuits reaches near 2×n! . This paper presents the first work to create all 3120218828 optimal 4-bit reversible circuits with up to 8 gates for the CNT (Controlled-NOT gate, NOT gate, and Toffoli gate) library, and it can quickly achieve 16 steps through specific cascading created circuits.
Optimal synthesis of reversible functions is a nontrivial problem. One of the major limiting factors in computing such circuits is the sheer number of reversible functions. Even restricting synthesis to 4-bit reversible functions results in a huge search space (16! ≈ 244functions). The output of such a search alone, counting only the space required to list Toffoli gates for every function, would require over 100 terabytes of storage. In this paper, we present two algorithms: one, that synthesizes an optimal circuit for any 4-bit reversible specification, and another that synthesizes all optimal implementations. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of all optimal 4-bit permutations, synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, and synthesis of existing benchmark functions; we compose a list of the hardest permutations to synthesize, and show distribution of optimal circuits. We further illustrate that our proposed approach may be extended to accommodate physical constraints via reporting LNN-optimal reversible circuits. Our results have important implications in the design and optimization of reversible and quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.