Realization and synthesis of reversible functions throng double-coset

Reversible circuits play an important role in quantum computation. The realization of reversible quantum circuits is required for any quantum computer to be universal. This paper¹ studies the realization and synthesis problem of reversible circuits. Using group theory, we present two sets of new 3-bit reversible logic gates, which included g₁g₂g₃ and nine complex gates, respectively. It is shown that any 3-bit reversible logic circuit is realizable by cascading NOT and Feynman gates, and at most one instance of our proposed gates. Given any n-bit reversible function, there are N distinct input patterns different from their corresponding outputs, where N ≤ 2ⁿ, and the other (2ⁿ - N) input patterns will be the same as their outputs. According to a synthesis example, it demonstrates that the main results are constructive and practical than others.