Networks for reversible logic

If we like to make an arbitrary permutation of a large number (say n) objects, where n is a non-prime number (n = pq, with both p and q integer), it is advantageous to arrange the objects in a rectangular p×q matrix. Then the permutation can be performed in three steps: first one applies a permutation where all objects remain in the same row, then one applies a permutation where all objects remain in the same column, and finally one applies a second permutation where all objects remain in the same row. In telecommunication, this remarkable theorem is the basis of so-called Clos networks, where w communication wires have to be permuted, according to one of the w! possible permutations. In binary digital communication, w wires transport one of the 2w possible messages. Reversible computing consists of applying a permutation, not to the w wires but to the 2w possible messages. The Clos approach allows us to build reversible binary computers very efficiently. The approach is somewhat less efficient for multiple-valued reversible logic and, unfortunately, is not applicable at all for arbitrary quantum circuits