 Some authors have questioned whether the mainstream quantum computing paradigm is general enough and, if not, whether some aspects of quantum mechanics may be exploited to design a quantum hypercomputer (Nielsen 1997, Calude and Pavlov 2002). The most prominent proposal for a quantum hypercomputer is due to Tien Kieu (2002, 2003, 2004, 2005). He argues that an appropriately constructed quantum system can decide whether an arbitrary Diophantine equation has an integral solution—a problem which is known to be unsolvable by TMs. Kieu’s method involves encoding a specific instance of the problem in an appropriate Hamiltonian, which represents the total energy of a quantum system. Kieu shows that such a system can dynamically evolve over time into an energy ground state that encodes the desired solution. Unfortunately, Kieu’s scheme does not appear to be workable. For one thing, it requires infinite precision in setting up and maintaining the system (Hodges 2005). For another, Kieu does not provide a successful criterion for knowing when the system has evolved into the solution state, and the problem of determining when the solution state is reached is Turing-uncomputable (Smith 2006b, Hagar and Korolev 2007). Thus, operating Kieu’s proposed hypercomputer would require already possessing hypercomputational powers.