On quantum hypercomputation

Quantum computation has attracted much attention and investment lately through its theoretical potential to speed up some important computations as compared to classical Turing computation. The most well-known and widely-studied form of quantum computation is the (standard) model of quantum circuits [1] which comprise several unitary quantum gates (belonging to a set of universal gates), arranged in a prescribed sequence to implement a quantum algorithm. Each quantum gate only needs to act singly on one qubit (quantum bit), a quantum generalisation of the classical bit, or on a pair of qubits. In view of this promising potential of quantum computation, another question then inevitably arises is whether the class of Turing computable functions can be extended by applying quantum principles. This kind of computation beyond the Turing barrier is also known as hypercomputation [2]. Initial efforts [3] have indicated that quantum computability is the same as classical computability. However, this negative conclusion is only valid for the standard quantum computation; but the standard model is not the only model of quantum computation. Nor does it necessarily exploit fully the principles of quantum mechanics. As an illustration of the ability of quantum mechanics, a truly random sequence of bits could be easily generated from a qubit initially prepared in a certain state, while Turing machines have to be content with only pseudo-random generators (see, for example, [4]). (Thus, it seems from the view point of Algorithmic Information Theory [5] that a finitely prepared qubit can have an infinite algorithmic informational theoretic complexity as compared to any finite Turing machine!) This ability to generate random numbers could thus be seen as a form of hypercomputation – albeit of some very limited application and of a nature which is non-harnessible or non-exploitable for universal computation. Also not being restricted by the standard model of quantum computation, we have claimed [6, 7, 8, 9, 10] to have a quantum algorithm for Hilbert’s tenth problem [11] despite the fact that the problem has been shown to be recursively noncomputable. The algorithm makes essential use of the Quantum Adiabatic Theorem (QAT) [12] and other results in the framework of Quantum Adiabatic Computation (QAC) [13], subject to some predetermined and arbitrarily small probability of error. We name this kind of algorithms Probabilistically Correct Algorithms to emphasise the fact that the end results from such algorithms are subject to some probabilities of being incorrect. Such error probabilities are necessary when there is, in principle, no other way to verify all the outcomes of the algorithms. We should note here that there is a whole hierarchy of the noncomputables [14]; that is, some are more ‘noncomputable’ than others. Computability of certain noncomputable, if could be ascertained, does not mean that all the noncomputables are then computable. Our claim of (quantum) computability is restricted in that sense, it is only applied to Hilbert’s tenth problem, or equivalently the Turing halting problem, or any equivalent problem. What constitutes, in particular, the noncomputability of Hilbert’s tenth problem? For each Diophantine equation without any parameter, there is nothing noncomputable about whether that particular equation has any integer solution or not. If we have not yet had a (recursive) procedure to obtain that answer then it does not mean that such a procedure is ever out of reach. Nor there is anything noncomputable about a collection of finitely many Diophantine equations, because we can always concatenate all the procedures for all the equations (as there is always a procedure in principle to determine the existence of solution for each equation) into a finitely collective procedure that can be applied to the whole collection. What constitutes the noncomputability of Hilbert’s tenth problem is the fact that we ask for