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

Multiple-valued reversible logic circuits

We consider the symmetric group S-n in the special case where n = pq (both p and q being integer). Applying Birkhoff's theorem, we prove that an arbitrary element of S-pq can be decomposed into a product of three permutations, the first and the third being elements of the Young subgroup S-p(q), whereas the second one is an element of the dual Young subgroup S-p(q). This leads to synthesis methods for arbitrary multiple-valued reversible logic circuits of logic width w. These circuits indeed form a group isomorphic to Sr-w, where r is the radix of the multiple-valued logic. A particularly efficient decomposition is found by choosing p = r and thus q = r(w-1). As a result, an arbitrary reversible logic circuit of radix r and width w is decomposed into a cascade of 2w - 1 control gates, i.e. logic building blocks, which manipulate only one of the w dits

Young subgroups for reversible computers

We consider the symmetric group S-n in the special case where n is composite: n = pq (both p and q being integer). Applying Birkhoff's theorem, we prove that an arbitrary element of S-pq can be decomposed into a product of three permutations, the first and the third being elements of the Young subgroup S-p(q), whereas the second one is an element of the dual Young subgroup S-p(q). This leads to synthesis methods for arbitrary reversible logic circuits of logic width w. These circuits form a group isomorphic to S(2)w. A particularly efficient synthesis is found by choosing p = 2 and thus q = 2(w-1). The approach illustrates a direct link between combinatorics, group theory, and reversible computing