Halting Probability Amplitude Quantum Computers
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The classical halting probability y i n troduced 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 a s F ermat's theorem. It contains the solution for the question of whether or not a particular exponential Diophantine equation has innnitely many o r a nite number of solutions. And, since is provable algorithmically incom-pressible," it is Martin-LL offChaitinnSolovay 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 pro-1 A t ypical realisation of C would be by an array of generalized four-port beam splitters 13. In what follows we shall assume that the program p i is coded classically. That is, we c hoose a nite code alphabet A and denote by A the set of all strings over A. A n y program p i is coded as a classical sequence p i = s 1i s 2i s ni 2 A , s ji 2 A. In what follows, p i will be abbreviated by p i. We assume preex coding 14, 1 , 1 5 , 33; i.e., the domain of C is preex-free such that no admissible program is the preex of another admissible program. Furthermore, without loss of generality, w e consider only empty input strings.
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