Pawel Kerntopf

A new heuristic algorithm for reversible logic synthesis

Reversible logic has applications in many fields, including quantum computing. Synthesis techniques for reversible circuits are not well developed, even for functions with a small number of inputs and outputs. This paper proposes an approach to reversible logic synthesis using a new complexity measure based on shared binary decision diagrams with complemented edges (instead of truth tables or PPRM forms, as in the previous algorithms). The approach can be used with arbitrary libraries of reversible logic gates and arbitrary cost functions. Experiments show promising results in comparison with the known approaches.

Multi-output Galois Field Sum of Products synthesis with new quantum cascades

Galois Field Sum of Products (GFSOP) leads to efficient multi-valued reversible circuit synthesis using quantum gates. In this paper, we propose a new generalization of ternary Toffoli gate and another new generalized reversible ternary gale with discussion of their quantum realizations. Algorithms for synthesizing ternary GFSOP using quantum cascades of these gates are proposed In both the synthesis methods, 5 ternary shift operators and ternary swap gate are used We also propose quantum realizations of 5 ternary shift operators and ternary swap gate. In the cascades of the new ternary gates, local mirrors, variable ordering, and product ordering techniques are used to reduce the circuit cost. Experimental results show that the cascade of the new ternary gates is more efficient than the cascade of ternary Toffoli gates.

On universality of general reversible multiple-valued logic gates

A set of p-valued logic gates (primitives) is called universal if an arbitrary p-valued logic function can be realized by a logic circuit built up from a finite number of gates belonging to this set. In this paper, we consider the problem of determining the number of universal single-gate libraries of p-valued reversible logic gates with two inputs and two outputs, under the assumption that constant signals can be applied to an arbitrary number of inputs. We have proved some properties of such gates and established that over 97% of ternary gates are universal.

Synthesis of multipurpose reversible logic gates

Regular reversible logic circuits, i.e. consisting of identical reversible gates, each of which is uniformly connected to its neighbors, have small garbage. Reversible gates realizing simultaneously some useful functions are very effective in synthesis of regular reversible logic circuits. In the paper we show how to create multipurpose reversible gates. Examples of efficient binary multipurpose reversible gates are also shown.

An approach to quantum cost optimization in reversible circuits

Recently, one of the main criteria used to evaluate reversible circuit designs is quantum cost. In this paper, an approach to reducing quantum cost of small-width reversible circuits is presented. Using our tool we have shown that for known benchmarks as well as designs taken from recent publications it is possible to obtain substantial savings in quantum cost (35% on average for 4-input benchmarks). It is also shown that quantum cost of 5-input circuits can be reduced using the same tool.

Synthesis of reversible circuits: A view on the state-of-the-art

Four main approaches to synthesis of reversible circuits are considered: cycle-based, transformation-based, ESOP-based and BDD-based as well as their advantages and disadvantages are discussed. It is indicated that the decisions in them are based on local information only what leads to very redundant designs. Suggestions for making global decisions are also presented. New directions of research are also described.

Optimal 4-bit Reversible Mixed-Polarity Toffoli Circuits

Optimal synthesis of reversible circuits is a very hard task. For example, up to year 2009 this problem had not been solved even for 4-bit reversible functions, in spite of intensive research during previous decade. In 2010, a method and a tool of practical usage for finding optimal circuits for any 4-bit reversible specification were finally developed. Namely, with sophisticated optimizations it was possible to find gate count optimal circuits for any 4-bit reversible function built from multi-control Toffoli gates. Last year, we published an extension to the algorithm, which allows to reduce the quantum cost of the resulting circuits. In this paper we present another extension to this approach. Namely, we have extended the reversible gate library to mixed-polarity multi-control Toffoli gates (i.e. with both positive and negative controls). Our experimental results for the known reversible benchmarks show that using mixed-polarity Toffoli gates gives significant savings in gate count. The paper presents results of different computational experiments including optimal 4-bit circuits for the known reversible benchmarks with respect to both gate count and quantum cost criteria.

A study of optimal 4-bit reversible circuit synthesis from mixed-polarity Toffoli gates

Optimal synthesis of reversible circuits is a very hard task. In 2010, a method and a tool of practical usage for finding optimal circuits built from multi-control Toffoli gates for any 4-bit reversible specification were finally developed. In 2011 we published an extension to the algorithm, which allows to reduce the quantum cost of the resulting circuits. In this paper we present another extension to this approach. Namely, we have extended the reversible gate library to mixed-polarity multi-control Toffoli gates (i.e. with both positive and negative controls). Our experimental results for the known reversible benchmarks show that using mixed-polarity Toffoli gates gives significant savings in gate count. The paper presents standard computational experiments for circuit synthesis. These results have implications in testing synthesis algorithms for reversible mixed-polarity circuits and quantum circuits.

Estimating the quality of complexity measures in heuristics for reversible logic synthesis

Reversible circuits have been intensively studied in recent years due to their applications in many areas, including quantum computing, nanotechnology and low-power design. Synthesis of reversible circuits differs significantly from the traditional logic synthesis. No satisfactory synthesis algorithm has been proposed for such circuits so far. All existing heuristic approaches to reversible logic synthesis utilize complexity measures. In this paper such complexity measures are analyzed and quantitative comparison of these approaches is presented. Possibilities for improving heuristic algorithms for reversible circuit synthesis are also discussed.

An approach to minimization of decision diagrams

One of the most promising concepts which has been developed for efficient representation functions is Linearly Transformed Binary Decision Diagram (LTBDD). We present extensions to LTBDDs called Function-driven Decision Diagrams (fDDs). The notion of fDDs is based on using simple balanced (including nonlinear) Boolean functions for defining transformations of decision diagrams. In this context a new scheme of preprocessing which corresponds to inverse transformations as well as using composition of transformations are very efficient for minimization of fDDs. The first experimental results show that fDDs driven by nonlinear Boolean functions can be more compact than LTBDDs, with a reasonable cost. Further extensions of fDDs are also mentioned such as Function-driven Kronecker Functional Decision Diagrams and Multiple-Valued Function-driven Decision Diagrams.