Halting probability amplitude of quantum computers

The classical halting probability introduced by Chaitin is generalized to quantum computations. Chaitin's [1, 2, 3] is a magic number. It is a measure for arbitrary programs to take a nite number of execution steps and then halt. It contains the solution for all halting problems, and hence to questions codable into halting problems, such as Fermat's theorem. It contains the solution for the question of whether or not a particular exponential Diophantine equation has in nitely many or a nite number of solutions. And, since is provable \algorithmically incompressible," it is Martin-Lof/Chaitin/Solovay random. Therefore, is both: a mathematicians \fair coin," and a formalist's nightmare. Here, is generalized to quantum computations. Consider a (not necessarily universal) quantum computer C and its ith program pi, which, at time t 2 Z, can be described by a quantum state [4, 5, 6, 7, 8, 9, 10, 11, 12]) jC(t; pi)i : (1) A typical realisation of C would be by an array of generalized four-port beam splitters [13]. In what follows we shall assume that the program pi is coded classically. That is, we choose a nite code alphabet A and denote by A the set of all strings over A. Any program pi is coded as a classical sequence #(pi) = s1is2i sni 2 A , sji 2 A. (In what follows, #(pi) will be abbreviated by pi.) We assume pre x coding [14, 1, 15, 3]; i.e., the domain of C is pre x-free such that no admissible program is the pre x of another admissible program. Furthermore, without loss of generality, we consider only empty input strings. The quantum omega was invented in a meeting of G. Chaitin, A. Zeilinger and the author (K. S.) in a Viennese co ee house (Caf e Braunerhof) in January 1991. Thus, the group should be credited for the original invention, whereas any blame should remain with the author. Journal of Universal Computer Science, vol. 1, no. 3 (1995), 201-204 submitted: 20/1/95, accepted: 27/2/95, appeared: 28/3/95, Springer Pub. Co.