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Design, Automation, and Test in Europe

Make it reversible: Efficient embedding of non-reversible functions

Alwin Zulehner, Robert Wille

Published Mar 1, 2017• 44 citations
Publisher

Abstract

Reversible computation became established as a promising concept due to its application in various areas like quantum computation, energy-aware circuits, and further areas. Unfortunately, most functions of interest are non-reversible. Therefore, a process called embedding has to be conducted to transform a non-reversible function into a reversible one - a coNP-hard problem. Existing solutions suffer from the resulting exponential complexity and, hence, are limited to rather small functions only. In this work, an approach is presented which tackles the problem in an entirely new fashion. We divide the embedding process into matrix operations, which can be conducted efficiently on a certain kind of decision diagram. Experiments show that improvements of several orders of magnitudes can be achieved using the proposed method. Moreover, for many benchmarks exact results can be obtained for the first time ever.

DOI
10.23919/date.2017.7927033

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Reversible computation became established as a promising concept due to its application in various areas like quantum computation, energy-aware circuits, and further areas. Unfortunately, most functions of interest are non-reversible. Therefore, a process called embedding has to be conducted to transform a non-reversible function into a reversible one - a coNP-hard problem. Existing solutions suffer from the resulting exponential complexity and, hence, are limited to rather small functions only. In this work, an approach is presented which tackles the problem in an entirely new fashion. We divide the embedding process into matrix operations, which can be conducted efficiently on a certain kind of decision diagram. Experiments show that improvements of several orders of magnitudes can be achieved using the proposed method. Moreover, for many benchmarks exact results can be obtained for the first time ever.
Reversible logic represents the basis for many emerging technologies and has recently been intensively studied. However, most of the Boolean functions of practical interest are irreversible and must be embedded into a reversible function before they can be synthesized. Thus far, an optimal embedding is guaranteed only for small functions, whereas a significant overhead results when large functions are considered. In this paper, we study this issue. We prove that determining an optimal embedding is coNP-hard already for restricted cases. Then, we propose heuristic and exact methods for determining both the number of additional lines as well as a corresponding embedding. For the approaches we considered sums of products and binary decision diagrams as function representations. Experimental evaluations show the applicability of the approaches for large functions. Consequently, the reversible embedding of large functions is enabled as a precursor to subsequent synthesis.
Reversible logic represents the basis for many emerging technologies and has recently been intensively studied. However, most of the Boolean functions of practical interest are irreversible and must be embedded into a reversible function before they can be synthesized. Thus far, an optimal embedding is guaranteed only for small functions, whereas a significant overhead results when large functions are considered. We study this issue in this article. We prove that determining an optimal embedding is coNP-hard already for restricted cases. Then, we propose heuristic and exact methods for determining both the number of additional lines and a corresponding embedding. For the approaches, we considered sum of products and binary decision diagrams as function representations. Experimental evaluations show the applicability of the approaches for large functions. Consequently, the reversible embedding of large functions is enabled as a precursor to subsequent synthesis.
Optimal synthesis of reversible functions is a non-trivial 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!~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 an algorithm that synthesizes an optimal circuit for any 4-bit reversible specification. We employ several techniques to make the problem tractable. We report results from several experiments, including synthesis of random 4-bit permutations, optimal synthesis of all 4-bit linear reversible circuits, synthesis of existing benchmark functions, and distribution of optimal circuits. Our results have important implications for the design and optimization of quantum circuits, testing circuit synthesis heuristics, and performing experiments in the area of quantum information processing.
Reversible logic studies have promising potential on energy lossless circuit design, quantum computation, nanotechnology, etc. Reversible logic features a one-to-one input output correspondence which makes the logic synthesis for reversible functions differs greatly from traditional Boolean functions. Exact synthesis methods can provide optimal solutions in terms of the total number of reversible gates in the synthesis results. Unfortunately, they may suffer from long computation time, due to the fact that the search space is likely to grow exponentially as the circuit size increases. Therefore, in this paper, we propose an efficient synthesis heuristic which provides high quality synthesis results of Toffoli network in more reasonable computation time. We use a weighted, directed graph for reversible function representation and complexity measurement. The proposed algorithm maximally decreases function complexity during synthesis steps. It has the ability to climb out of local minimums and guarantees algorithm convergence. The experimental results show that our algorithm can achieve optimal or very close to optimal solutions with computation time several orders of magnitude less than the exact methods. Compared with other heuristics, our method demonstrates superior performance in terms of reversible gate count as well as computation time.
Quantum computing promises substantial speedups by exploiting quantum mechanical phenomena such as superposition and entanglement. Corresponding design methods require efficient means of representation and manipulation of quantum functionality. In the classical domain, decision diagrams have been successfully employed as a powerful alternative to straightforward means such as truth tables. This motivated extensive research on whether decision diagrams provide similar potential in the quantum domain-resulting in new types of decision diagrams capable of substantially reducing the complexity of representing quantum states and functionality. From an implementation perspective, many concepts and techniques from the classical domain can be re-used in order to implement decision diagrams packages for the quantum realm. However, new problems-namely how to efficiently handle complex numbers-arise. In this work, we propose a solution to overcome these problems. Experimental evaluations confirm that this yields improvements of orders of magnitude in the runtime needed to create and to utilize these decision diagrams. The resulting implementation is publicly available as a quantum DD package at http://iic.jku.at/eda/research/quantum_dd.
Reversible logic studies have promising potential on energy lossless circuit design, quantum computation, nanotechnology, etc. Reversible logic features a one-to-one input output correspondence which makes the logic synthesis for reversible functions differs greatly from traditional Boolean functions. Exact synthesis methods can provide optimal solutions in terms of the total number of reversible gates in the synthesis results. Unfortunately, they may suffer from long computation time, due to the fact that the search space is likely to grow exponentially as the circuit size increases. Therefore, in this paper, we propose an efficient synthesis heuristic which provides high quality synthesis results of Toffoli network in more reasonable computation time. We use a weighted, directed graph for reversible function representation and complexity measurement. The proposed algorithm maximally decreases function complexity during synthesis steps. It has the ability to climb out of local minimums and guarantees algorithm convergence. The experimental results show that our algorithm can achieve optimal or very close to optimal solutions with computation time several orders of magnitude less than the exact methods. Compared with other heuristics, our method demonstrates superior performance in terms of reversible gate count as well as computation time.