Group Theory Based Synthesis of Binary Reversible Circuits

Abstract

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’-Controlled-Not gate), where the numbers of Toffoli and NOT gates required in the realization are bounded by $(n + \lfloor \frac{n}{3} \rfloor)(3 \times 2^{2n-3}-2^{n+2})$ and $4n(n + \lfloor \frac{n}{3} \rfloor)2^n$, respectively. A provable constructive synthesis algorithm is derived. The time complexity of the algorithm is $\frac{10}{3}n^2 \cdot 2^n$. Our algorithm is exponentially lower than breadth-first search based synthesis algorithms with respect to space and time complexities.