Group Theory Based Synthesis of Binary Reversible Circuits

This paper presents an important result addressing a fundamental question in synthesizing binary reversible logic circuits for quantum computation. We show that any even-reversible-circuit of n (n > 3) qubits can be realized using NOT gate and Toffoli gate (‘2’-ControlledNot gate), where the numbers of Toffoli and NOT gates required in the realization are bounded by (n+ n3 )(3×2 2n−3−2n+2) and 4n(n+ n3 )2 , respectively. A provable constructive synthesis algorithm is derived. The time complexity of the algorithm is 10 3 n 2 · 2. Our algorithm is exponentially lower than breadth-first search based synthesis algorithms with respect to space and time complexities.