Decomposition of a Linear Reversible Computer: Digital Versus Analog
Linear reversible transformations in the Galois field GF(2) and linear reversible transformations in the field of real numbers show both resemblances and differences. The former constitute a finite group isomorphic to the general linear group GL(w, 2), the latter constitute an infinite, i.e. Lie, group isomorphic to the general linear group GL(w, R) (where w is the logic width of the computation, i.e. respectively the number of bits and the number of real numbers, processed by the computer). Generators of the former group consist of merely control gates; generators of the latter group consist of both control gates and scale gates.
reversible computing group theory lifting scheme