This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.

Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. “To believe otherwise is merely to cling to a prejudice which only gives rise to further prejudices... ” 1

We give an update on a quantum adiabatic algorithm for the Turing noncomputable Hilbert’s tenth problem, and briefly go over some relevant issues and misleading objections to the algorithm. Keywords: Quantum adiabatic computation; Turing computability; Hilbert’s tenth problem 1.

Quantum hypercomputation based on the dynamical algebra su(1, 1)