Majority-based reversible logic gates

Reversible logic plays an important role in the synthesis of circuits for quantum computing. In this paper, we introduce families of reversible gates based on the majority Boolean function (MBF) and we prove their properties in reversible circuit synthesis. These gates can be used to synthesize reversible circuits of minimum “scratchpad register width” for arbitrary reversible functions. We show that, given a MBF f with $2k+1$ inputs, f can be implemented by a reversible logic gate with $2k+1$ inputs and $2k+1$ outputs, i.e., without any constant inputs. We also demonstrate new gates from this family with very efficient quantum realizations for majority-based applications. They can be used to synthesize any reversible function of the same width in conjunction with inverters and Feynman (2-qubit controlled-NOT) gates. The gate universality problem is formulated in terms of elementary group theory and solved using the algebraic software GAP.