The increased effort in recent years towards methods for computer aided design of reversible logic circuits has also lead to research in algorithms for optimising the resulting circuits; both with higher-level data structures and directly on the reversible circuits. To obtain structural patterns that can be replaced by a cheaper realisation, many direct algorithms apply so-called moving rules; a simple form of rewrite rules that can only swap gate order.

In this paper we first describe the few basic rules that are needed to perform rewriting directly on reversible logic circuits made from general Toffoli circuits. We also show how to use these rules to derive more complex formulas. The major difference compared to existing approaches is the use of negative controls (white dots), which significantly increases the algebraic strength. We show how existing optimisation approaches can be adapted as problems based on our rewrite rules.

Finally, we outline a path to generalising the rewrite rules by showing their forms for reversible control-gates. This can be used to expand our method to other gates such as the controlled-swap gate or quantum gates.

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White dots do matter: rewriting reversible logic circuits

Real quantum computing technologies have different restrictions and constraints which need to be considered during circuit synthesis. In certain technologies, only physically adjacent qubits can interact, which restricts their realizations to only linear nearest neighbor (LNN) architecture. In this work, we formulate the line ordering problem in LNN architecture as task assignment problem to find a mapping (permutation) between task graph and processor graph with minimum cost. We propose two different approaches, a greedy heuristic and a meta-heuristic algorithm based on Harmony Search to solve the task assignment problem. Experimental results show that our algorithms were able to reduce the quantum cost of benchmark circuits by approximately 30 % on average. Moreover, the proposed algorithms were compared to one recently proposed ordering algorithm and were found to further improve the cost by approximately 16 %.

Line ordering of reversible circuits for linear nearest neighbor realization

We have defined a new method for automatic construction of reversible logic circuits by using the genetic programming approach. The choice of the gate library is 100% dynamic. The algorithm is capable of accepting all possible combinations of the following gate types: NOT TOFFOLI, NOT PERES, NOT CNOT TOFFOLI, NOT CNOT SWAP FREDKIN, NOT CNOT TOFFOLI SWAP FREDKIN, NOT CNOT PERES, NOT CNOT SWAP FREDKIN PERES, NOT CNOT TOFFOLI PERES and NOT CNOT TOFFOLI SWAP FREDKIN PERES. Our method produced near optimum circuits in some cases when a particular subset of gate types was used in the library. Meanwhile, in some cases, optimal circuits were produced due to the heuristic nature of the algorithm. We compared the outcomes of our method with several existing synthesis methods, and it was shown that our algorithm performed relatively well compared to the previous synthesis methods in terms of the output efficiency of the algorithm and execution time as well.

Reversible circuit synthesis by genetic programming using dynamic gate libraries

NOT, Feynman, and Toffoli gates are the practical building blocks of reversible circuit synthesis techniques. NOT and Feynman gates are primitive quantum gates. Toffoli gates are macro-level gates and realized using various primitive quantum gates. In reversible circuit design papers, logic level circuits are first realized using NOT, Feynman, and Toffoli gates and then the Toffoli gates are decomposed into primitive quantum gates. However, realizations of reversible circuits using electronic devices are not given adequate emphasis. In this paper, we propose preliminary ideas of implementations of NOT, Feynman, and Toffoli gates using single-electron transistor (SET), which is a nano-electronic quantum-tunneling device. NOT gate implementation requires one SET and one pull-up resistor. Feynman gate implementation requires two SETs and two pull-up resistors. Three-input Toffoli gate implementation requires three SETs and two pull-up resistors. The average static power consumptions of NOT, Feynman, and three-input Toffoli gate realizations are 0.091 nW, 0.497 nW, and 0.504 nW, respectively. The proposed implementations will open a facet of nano-electronic low-power realizations of reversible circuits.

Single-Electron Transistor Based Implementation of NOT, Feynman, and Toffoli Gates

The paper discusses various applications of permutation group theory in the synthesis of reversible logic circuits consisting of Toffoli gates with negative control lines. An asymptotically optimal synthesis algorithm for circuits consisting of gates from the NCT library is described. An algorithm for gate complexity reduction, based on equivalent replacements of gates compositions, is introduced. A new approach for combining a group-theory-based synthesis algorithm with a Reed-Muller-spectra-based synthesis algorithm is described. Experimental results are presented to show that the proposed synthesis techniques allow a reduction in input lines count, gate complexity or quantum cost of reversible circuits for various benchmark functions.

Zamani

Papers

Rule-based optimization of reversible circuits