The natural Hilbert Space of quantum particles can implement maximum-likelihood (ML) decoding of classical information. The 'Quantum Product Algorithm' (QPA) is computed on a Factor Graph, where function nodes are unitary matrix operations followed by appropriate quantum measurement. QPA is like the Sum-Product Algorithm (SPA), but without summary, giving optimal decode with exponentially finer detail than achievable using SPA. Graph cycles have no effect on QPA performance. QPA must be repeated a number of times before successful and the ML codeword is obtained only after repeated quantum 'experiments'. ML amplification improves decoding accuracy, and Distributed QPA facilitates successful evolution.

Minor modifications. 24 pages, Latex, 14 figures, Presented in part at 2nd Int. Symp. on Turbo Codes and Related Topics, Brest, France, Sept 4-7, 2000 Accepted for publication in "Annals of Telecom." 2001