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Journal of Engineering and Sustainable Development

Complexity Analysis for Garbage-Free Reversible Computation of Boolean Functions with Variable Ancilla

Zaid Al-Wardi

Published Nov 1, 2026• 0 citations
Publisher

Abstract

Reversible computation is a fundamental concept in quantum circuits and low-power design. As the demand for powerful and energy-efficient computing grows, novel design methods for reversible circuits become critical for achieving high performance with low power consumption. Current reversible circuit synthesis methodologies fall into two classes, each minimizing a different parameter in circuit complexity, but often result in relatively high values in another parameter. Post-synthesis optimizations moderate complexity parameters, basically by trading off between them. However, these optimizations depend on specific gate patterns and may not always meet design constraints effectively. This paper proposes a novel recursive decomposition scheme for large Boolean functions using a single-target gate model in reversible circuit synthesis. The proposed scheme systematically and efficiently manages the trade-off between reversible circuit complexity measures, offering gradually varying parameters at each level in the recursion. The scalability of the proposed approach is presented as a comprehensive complexity analysis for the mathematical model. Rather than providing a globally optimal solution, this approach offers a range of solutions, allowing designers to choose the most suitable option based on specific design constraints.

DOI
10.31272/jeasd.2923

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Reversible computation is a fundamental concept in quantum circuits and low-power design. As the demand for powerful and energy-efficient computing grows, novel design methods for reversible circuits become critical for achieving high performance with low power consumption. Current reversible circuit synthesis methodologies fall into two classes, each minimizing a different parameter in circuit complexity, but often result in relatively high values in another parameter. Post-synthesis optimizations moderate complexity parameters, basically by trading off between them. However, these optimizations depend on specific gate patterns and may not always meet design constraints effectively. This paper proposes a novel recursive decomposition scheme for large Boolean functions using a single-target gate model in reversible circuit synthesis. The proposed scheme systematically and efficiently manages the trade-off between reversible circuit complexity measures, offering gradually varying parameters at each level in the recursion. The scalability of the proposed approach is presented as a comprehensive complexity analysis for the mathematical model. Rather than providing a globally optimal solution, this approach offers a range of solutions, allowing designers to choose the most suitable option based on specific design constraints.
Reversible computation is gaining increasing relevance in the context of several post-CMOS technologies, the most prominent of those being Quantum computing. One of the key theoretical problem pertaining to reversible logic synthesis is the upper bound of the gate count. Compared to the known bounds, the results obtained by optimal synthesis methods are significantly less. In this paper, we connect this problem with the multiplicative complexity analysis of classical Boolean functions. We explore the possibility of relaxing the ancilla and if that approach makes the upper bound tighter. Our results are negative. The ancilla-free synthesis methods by using transformations and by starting from an Exclusive Sum-of-Product (ESOP) formulation remain, theoretically, the synthesis methods for achieving least gate count for the cases where the number of variables nnn is <8< 8<8 and otherwise, respectively.
Reversible logic circuit is a necessary construction for achieving ultra low power dissipation as well as for prominent post-CMOS computing technologies such as Quantum computing. Consequently automatic synthesis of a Boolean function using elementary reversible logic gates has received significant research attention in recent times, creating the domain of reversible logic synthesis. In this paper, we study the complexity of reversible logic synthesis. The problem is separately studied for bounded-ancilla and ancilla-free optimal synthesis approaches. The computational complexity for both cases are linked to known/presumed hard problems. Finally, experiments are performed with a shortest-path based reversible logic synthesis approach and a (0-1) ILP-based formulation.
Emerging technologies with asymptotic zero power dissipation, such as quantum computing, require the logical operations to be done in a reversible manner. In recent years, the problem of synthesizing Boolean functions in the reversible logic domain has gained significant research attention. The efficiency of the synthesis methods is measured in terms of quantum cost, gate cost, garbage lines, logic depth and speed of synthesis. In this paper, we present an approach based on Exclusive sum-of-Products (ESOP), which allows the user to explore the trade-off between quantum cost and garbage lines. The proposed technique adds a new dimension to the reversible logic synthesis solutions. We demonstrate by detailed experiments that controlled improvement in quantum cost and gate count by increasing garbage count can be achieved. In some cases, improved quantum cost and gate count compared to state-of-the-art synthesis methods are reported. Furthermore, we propose a novel rule-based approach to achieve ancilla-free reversible logic synthesis starting from an ESOP formulation.
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.
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.
Reversible logic has emerged as a pivotal area of study in the realm of computing, primarily due to its potential for energy efficient computation. Reversible cryptographic logic is an important area in applications such as signal processing, nanotechnology, bio-information, low power and lightweight applications and cyber security. The area and power constraints are as well taken into account of cryptographic protocol. In addition, intruders and hackers may snoop or modify identified data sent over the channel. Reversible cryptographic logic can be installed to prevent this situation. For this there is a need to study with optimal metrics viz. quantum cost, area, and performance. The feedback shift register is used for encryption and decryption techniques in a symmetrical way. This paper presents a comprehensive mathematical modeling framework for reversible logic circuits, focusing on the fundamental principles that govern their operation. We explore various reversible gate designs, analyze their operational efficiencies, and derive mathematical expressions that describe their behavior. By employing tools from combinatorial optimization and linear algebra, we investigate the trade-offs between circuit complexity and energy consumption. The modeling framework is validated through simulation results, which illustrate the advantages of reversible logic over traditional irreversible circuits in specific applications. Our findings highlight the significance of reversible logic in advancing quantum computing, low power digital systems and sustainable computing paradigms.