Quantum Iterative Deepening with an application to the Halting problem

TL;DR

This paper introduces a quantum iterative deepening approach that enables halting detection without disrupting computation, allowing for non-terminating processes and potential speedups, with applications to classical Turing machine simulation.

Contribution

It proposes a novel quantum computation method combining production systems and Grover's algorithm to address halting detection and undecidable problems.

Findings

Halting detection without interference in quantum computation

Supports non-terminating quantum processes

Potential for speedup in parallelizable quantum algorithms

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 signalling 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 \textit{Entscheidungsproblem} which require universal computational models, \textit{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.