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Fredkin/Toffoli Templates for Reversible Logic Synthesis

Dmitri Maslov, Gerhard W. Dueck, D. Michael Miller

Published Jan 1, 2003• 44 citations
OpenAlex

Abstract

Abstract—Reversible logic has applications in quantum com-puting, low power CMOS, nanotechnology, optical computing, and DNA computing. The most common reversible gates are the Toffoli gate (which performs a controlled NOT) and the Fredkin gate (performs a controlled swap of two inputs). Our synthesis algorithm first finds a cascade of Toffoli and Fredkin gates with no backtracking and minimal look-ahead. Next we apply trans-formations that reduce the size of the circuit. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation that reduces the gate count can be applied. In this paper we show that Toffoli and Fredkin gates behave in a similar manner. Therefore, some gates in the templates may not need to be specified—they can match a Toffoli or a Fredkin gate. We formalize this by introducing the box gate. All templates with less than six gates are enumerated and classified. We synthesize all three input, three output reversible functions and compare our results to those obtained previously. I.

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Abstract—Reversible logic has applications in quantum com-puting, low power CMOS, nanotechnology, optical computing, and DNA computing. The most common reversible gates are the Toffoli gate (which performs a controlled NOT) and the Fredkin gate (performs a controlled swap of two inputs). Our synthesis algorithm first finds a cascade of Toffoli and Fredkin gates with no backtracking and minimal look-ahead. Next we apply trans-formations that reduce the size of the circuit. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation that reduces the gate count can be applied. In this paper we show that Toffoli and Fredkin gates behave in a similar manner. Therefore, some gates in the templates may not need to be specified—they can match a Toffoli or a Fredkin gate. We formalize this by introducing the box gate. All templates with less than six gates are enumerated and classified. We synthesize all three input, three output reversible functions and compare our results to those obtained previously. I.
Reversible logic has applications in quantum computing, low power CMOS, nanotechnology, optical computing, and DNA computing. The most common reversible gates are the Toffoli gate and the Fredkin gate. Our synthesis algorithm first finds a cascade of Toffoli and Fredkin gates with no backtracking and minimal look-ahead. Next we apply transformations that reduce the size of the circuit. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation that reduces the gate count can be applied. In this paper we show that Toffoli and Fredkin gates behave in a similar manner. Therefore, some gates in the templates may not need to be specified-they can match a Toffoli or a Fredkin gate. We formalize this by introducing the box gate. All templates with less than six gates are enumerated and classified. We synthesize all three input, three output reversible functions and compare our results to those obtained previously.
Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. This paper addresses the above synthesis approach. We present a basic method and, based on that, a bidirectional synthesis algorithm which produces a network of Toffoli gates realizing a given reversible specification. An asymptotically optimal modification of the basic synthesis algorithm employing generalized mEXOR gates is also presented. Transformations are then applied using template matching. The basis for a template is a network of gates that realizes the identity function. If a sequence of gates in the synthesized network matches a sequence comprised of more than half the gates in a template, then a transformation using the remaining gates in the template can be applied resulting in a reduction in the gate count for the synthesized network. All templates with up to six gates are described in this paper. Experimental results including an exhaustive examination of all 3-variable reversible functions and a collection of benchmark problems are presented. The paper concludes with suggestions for further research.
Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. This paper addresses the second step. Transformations are accomplished via template matching. The basis for a template is a network with Ñ gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation reducing the gate count can be applied. All templates for Ñ � � are described in this paper. 1
Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Tooli networks can be divided into two steps. First, nd a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. This paper addresses the second step. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation reducing the gate count can be applied. All templates for m 7a re described in this paper.
Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired junction. Second, transform the network such that it uses fewer gates, while realizing the same function. This paper addresses the second step. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation reducing the gate count can be applied. All templates for m/spl les/7 are described in this paper.
The research in reversible logic has been attracted in recent years. It has applications in quantum computing, low-power CMOS design and nanotechnology. Many methods on how to cascade a reversible network have been proposed. And all these networks are composed of Toffoli/Fredkin gates. In this paper, we present a method that synthesizes a network with family of Positive/Negative Control (PNC) gates. Through introducing some examples for 3×3 reversible functions, and comparing with the network which is composed of Toffoli gates, the results show that introducing the PNC gates can produce an improvement for the number of gates in the reversible network, and have an advantage in reducing the cost of network.