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Templates for Toffoli Network Synthesis

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

Published Jan 1, 2003• 11 citations
OpenAlex

Abstract

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.

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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.

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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 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 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.
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.
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.
A reversible logic function maps each of the 2m input patterns to a unique output pattern. The network cascade structure problem is to realize a reversible function by a cascade of primitive reversible gates. We present a method that synthesizes a network with family of Toffoli gates. The cascade algorithm finds cascade of Toffoli gates with no backtracking and minimal look-ahead, and apply transformations that reduce the number of gates in the network. We have synthesized all three input and output reversible functions and compare our results to the other optimal results. We also present the results of applying our synthesis method to obtain networks for number of benchmark function.