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Issue No.09  Sept. (2012 vol.61)
pp: 13411353
Oleg Golubitsky , Google, Inc., Waterloo
Dmitri Maslov , University of Waterloo, Waterloo
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2011.144
ABSTRACT
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 4bit reversible functions results in a huge search space (16! \approx 2^{44} functions). 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 4bit 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 4bit permutations, synthesis of random 4bit permutations, optimal synthesis of all 4bit 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 LNNoptimal 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.
INDEX TERMS
Reversible circuits, logic synthesis, quantum circuits.
CITATION
Oleg Golubitsky, Dmitri Maslov, "A Study of Optimal 4Bit Reversible Toffoli Circuits and Their Synthesis", IEEE Transactions on Computers, vol.61, no. 9, pp. 13411353, Sept. 2012, doi:10.1109/TC.2011.144
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