The problem of synthesis and optimisation of logical reversible and quantum circuits from functional descriptions represented as decision diagrams is considered. It is one of the key problems being solved with the aim of creating quantum computing technology and quantum computers. A new method of stepwise transformation of the initial functional specification to a quantum circuit is proposed, which provides for the following project states: reduced ordered binary decision diagram, if-decision diagram, functional if-decision diagram, reversible circuit and quantum circuit. The novelty of the method consists in extending the Shannon and Davio expansions of a Boolean function on a single variable to the expansions of the same Boolean function on another function with obtaining decomposition products that are represented by incompletely defined Boolean functions. Uncertainty in the decomposition products gives remarkable opportunities for minimising the graph representation of the specified function. Instead of two outgoing branches of the binary diagram vertex, three outgoing branches of the if-diagram vertex are generated, which increase the level of parallelism in reversible and quantum circuits. For each transformation step, appropriate mapping rules are proposed that reduce the number of lines, gates and the depth of the reversible and quantum circuit. The comparison of new results with the results given by the known method of mapping the vertices of binary decision diagram into cascades of reversible and quantum gates shows a significant improvement in the quality of quantum circuits that are synthesised by the proposed method.
The problem of synthesis and optimisation of logical reversible and quantum circuits from functional descriptions represented as decision diagrams is considered. It is one of the key problems being solved with the aim of creating quantum computing technology and quantum computers. A new method of stepwise transformation of the initial functional specification to a quantum circuit is proposed, which provides for the following project states: reduced ordered binary decision diagram, if-decision diagram, functional if-decision diagram, reversible circuit and quantum circuit. The novelty of the method consists in extending the Shannon and Davio expansions of a Boolean function on a single variable to the expansions of the same Boolean function on another function with obtaining decomposition products that are represented by incompletely defined Boolean functions. Uncertainty in the decomposition products gives remarkable opportunities for minimising the graph representation of the specified function. Instead of two outgoing branches of the binary diagram vertex, three outgoing branches of the if-diagram vertex are generated, which increase the level of parallelism in reversible and quantum circuits. For each transformation step, appropriate mapping rules are proposed that reduce the number of lines, gates and the depth of the reversible and quantum circuit. The comparison of new results with the results given by the known method of mapping the vertices of binary decision diagram into cascades of reversible and quantum gates shows a significant improvement in the quality of quantum circuits that are synthesised by the proposed method.
Abstract Reversible circuit synthesis methods based on decision diagrams achieve low quantum costs but do not account for quantum bit (qubit) limits for the application of reversible logic in quantum computing. Here, a synthesis method using sub-graphs of shared functional decision diagrams (SFDDs) is proposed for reducing the number of lines when synthesizing reversible circuits. An SFDD is partitioned into sub-graphs by exploiting the longest dominant-active paths, and the sub-graphs are mapped to reversible gate cascades. To further reduce the number of lines, template root matching is presented for reusing circuit lines. Experimental results indicate that the proposed method achieves the known minimum number of lines in many cases and has good scalability. Although the proposed method increases the quantum cost over a prior method based on functional decision diagrams, it significantly reduces the number of lines in most cases. Compared with the one-pass method using quantum multiple-valued decision diagrams, the proposed method reduces the quantum cost without increasing the number of lines in many cases. When compared with the lookup table-based method using a direct mapping flow, the method reduces the number of lines in a few cases. Thus, the method aids in the physical realization of a quantum circuit.
Reversible logic became a promising alternative to traditional circuits because of its applications in emerging technologies such as quantum computing, low-power design, DNA computing, or nanotechnologies. As a result, synthesis of the respective circuits is an intensely studied topic. However, most synthesis methods are limited, because they rely on a truth table representation of the function to be synthesized. In this paper, the authors present a synthesis approach that is based on Binary Decision Diagrams (BDDs). The authors propose a technique to derive reversible or quantum circuits from BDDs by substituting all nodes of the BDD with a cascade of Toffoli or quantum gates, respectively. Boolean functions containing more than a hundred of variables can efficiently be synthesized. More precisely, a circuit can be obtained from a given BDD using an algorithm with linear worst case behavior regarding run-time and space requirements. Furthermore, using the proposed approach, theoretical results known from BDDs can be transferred to reversible circuits. Experiments show better results (with respect to the circuit cost) and a significantly better scalability in comparison to previous synthesis approaches.
Reversible logic is a key technique for quantum computing so leading to low-power designs. However, current synthesis algorithms for reversible circuits are low efficiency and do not obtain optimized reversible circuits, so they are only applied to small logic functions. In this paper, we propose a new method based on positive Davio expansion to synthesize reversible circuits, which generates a positive Davio decision diagram for a logic function and transfers diagram nodes to reversible circuits. The algorithm has advantages of optimizing area and fast synthesis speed compared to BDD (Binary decision diagram) based and RM (Reed-Muller) based synthesis method, so it can be adapted for large functions.
With the continuous shrinkage of transistor sizes in very large scale integrated circuits, power consumption forms a serious concern to be tackled. With their ability to allow for zero energy dissipation, reversible circuits have been considered as promising solution to meetup with low power design requirements. Moreover, advances in their synthesis methodologies can be easily applied to quantum circuits due to the inherited reversibility of the latter. Although numerous algorithms have been proposed in the literature to synthesize reversible circuits and map them into their corresponding quantum circuits, the scalability and computational effort of such algorithms form a serious concern when synthesizing large size input functions. Binary Decision Diagram-based synthesis for reversible circuits has shown great evidence in realizing reversible circuits with low quantum cost through exploiting proper reduction rules for smaller graph size. However, the order of the variables in the decision diagram impacts its overall size, and thus, the cost of its corresponding reversible circuit. While several reordering algorithms have been proposed in this manner, their direct impact on the quantum cost has not been considered. In this article, a Binary Decision Diagram-based algorithm for reversible circuit synthesis is proposed to synthesize reversible circuits for a given Boolean function with low quantum cost through exploiting a linearized relationship between the decision diagram size and the corresponding quantum cost. Thereafter, different decision diagram reordering algorithms have been integrated with the proposed algorithm and compared in terms of their impact on the quantum cost. Experimental results show that Genetic Algorithm-based reordering for decision diagram, supported with, cycle crossover, inverse mutation, and tournament selection, results in the least quantum cost of the output circuit if compared with other algorithms due to its property in preserving the nodes of the decision diagram in their near-optimal locations during the optimiation recipe.
Synthesis of reversible and quantum logic has become an intensely studied topic in the last years. However, most synthesis methods are limited, since they rely on a truth table representation of the function to be synthesized. BDD-based synthesis offers an alternative. Here, reversible or quantum circuits are derived from a function given as Binary Decision Diagram (BDD) by substituting all nodes of the BDD with a cascade of Toffoli or elementary quantum gates, respectively. As a result, the application of the approach is not limited by the truth table of the function but by the (quite more efficient) BDD representation. Furthermore, many optimization techniques for BDDs exist which can be exploited. In this work, we evaluate the effect of three optimization methods for BDDs (namely shared nodes, complement edges, and advanced orderings) on the resulting reversible and quantum circuits. We describe in detail the adjustments, which have to be done to support these optimizations for synthesis, and discuss possible improvements and drawbacks. In a case study, the effects are experimentally evaluated. The results showed, that applying these optimization techniques leads to significant smaller circuits (with respect to number of gates and lines) in most of the cases.
Reversible circuits are an emerging technology where all computations are performed in an invertible manner. Motivated by their promising applications, e.g. in the domain of quantum computation or in the low-power design, the synthesis of such circuits has been intensely studied. However, how to automatically realize reversible circuits with the minimal number of lines for large functions is an open research problem. In this paper, we propose a new synthesis approach which relies on concepts that are complementary to existing ones. While “conventional” function representations have been applied for synthesis so far (such as truth tables, ESOPs, BDDs), we exploit Quantum Multiple-valued Decision Diagrams (QMDDs) for this purpose. An algorithm is presented that performs transformations on this data-structure eventually leading to the desired circuit. Experimental results show the novelty of the proposed approach through enabling automatic synthesis of large reversible functions with the minimal number of circuit lines. Furthermore, the quantum cost of the resulting circuits is reduced by 50% on average compared to an existing state-of-the-art synthesis method.