Abstract
Classical models of computation traditionally resort to halting schemes in order to enquire about the state of a computation. In such schemes, a computational process is responsible for signaling an end of a calculation by setting a halt bit, which needs to be systematically checked by an observer. The capacity of quantum computational models to operate on a superposition of states requires an alternative approach. From a quantum perspective, any measurement of an equivalent halt qubit would have the potential to inherently interfere with the computation by provoking a random collapse amongst the states. This issue is exacerbated by undecidable problems such as the Entscheidungsproblem which require universal computational models, e.g. the classical Turing machine, to be able to proceed indefinitely. In this work we present an alternative view of quantum computation based on production system theory in conjunction with Grover's amplitude amplification scheme that allows for (1) a detection of halt states without interfering with the final result of a computation; (2) the possibility of non-terminating computation and (3) an inherent speedup to occur during computations susceptible of parallelization. We discuss how such a strategy can be employed in order to simulate classical Turing machines.
Highlights
The status of any computation can be determined through a halt state
In its seminal paper [6], Deutsch emphasizes that a quantum computer needs the ability to operate on an input that is a superposition of computational basis in order to be ‘‘fully quantum’’, When confronted with the halting issue Myers naturally raised the question if a universal quantum computer could ever be fully quantum? And how would such a computational model eventually function? We aim to provide an answer to these questions by developing an alternative proposal to quantum Turing machines based on production system theory
As Miyadera stated, the notion of probabilistic halting in the context of quantum Turing machines cannot be avoided, suggesting that the standard halting scheme of traditional quantum computational models needs to be reexamined [14]
Summary
The status of any computation can be determined through a halt state. The concept of the halting state has some important subtleties in the context of quantum computation. Ekert draws attention to this fact stating that there are two possibilities to circumvent such an issue, namely [1]: either run the computation for some predetermined number of steps or alternatively employ a halt flag. This flag is employed by a computational model to signal an end of the calculation. Such a flag is represented by a halt bit which is initialized to 0 and set to 1 once the computation terminates. Determining if a computation has finished is a matter of checking if the halt bit is set to 1, a task that can be accomplished through some form of periodic observation
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