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The authors have declared that no competing interests exist.

Wrote the paper: LT AW.

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

Classically, 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. The first one of these relates to quantum state evolution which needs to be expressed through unitary operators that represent reversible mappings. As a consequence, two successive states cannot be equal. Ekert draws attention to this fact stating that there are two possibilities to circumvent such an issue, namely

Furthermore, undecidable problems, such as the famous

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More generally, suppose that the compound system after the unitary evolution

Consequently, observing the halt qubit after

Ideally, one could argue that any von Neumann measurement should only be performed after all parallel computations have terminated. Indeed, some problems may allow one to determine

Following Myers observation of the conflict between quantum computation and system observation a number of authors provided meaningful contributions to the question of halting in quantum Turing machines. Ozawa

Measurement-based quantum Turing machines as a model for computation were defined in

In

In its seminal paper

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

With this work we are particularly interested in: (1) preserving the original principles proposed by Deutsch of linearity and unitary operators, in contrast with other proposals such as

The rest of this introduction presents an overview of the main concepts required for a complete understanding of the results that will be presented, namely: (1) Subsection

Our approach to the detection of quantum halting states requires fixing a computational model. This step is required since our proposal depends on the set of state transitions occurring during a computational process. We choose not to focus on Turing machines, instead our proposal will be formulated in terms of production system theory. This decision is based on the fact that the quantum Turing machine model was already well explored by Deutsch

Production system theory is also well suited to support tree search, a form of graph search from which we drew our initial inspiration. In addition, the classical counterparts of both models were shown to be equivalent in computational power

Let

The working memory

The set of production rules

Each rules precondition is matched against the contents of the working memory. If the precondition is met then the action part of the rule can be applied, changing the contents of the working memory.

The tuple

The control system

Let

With these definitions in mind it becomes possible to develop a suitable model for a quantum production system. Namely, the complex valued control strategy would need to behave as illustrated in Expression 6 where

The amplitude value provided would also have to be in accordance with Expression 7,

We will employ the notation described in

Universal models of computation are capable of calculating

The unbounded minimization operator can be perceived as a computational procedure responsible for repeatedly evaluating a function with different inputs

From a quantum computation perspective, it is possible to perform a generic search for solution states through amplitude amplification schemes such as the one described by Grover in

The quantum search algorithm employs an oracle

It is important to mention that we employed some care when defining the oracle in terms of registers

Grover's algorithm starts by setting up a superposition of

In addition it is also possible that the space includes several solutions. Accordingly, let

Is it possible to present an adequate mapping of our quantum production system that is suitable to be applied alongside Grover's algorithm? A comparison of Expression 6 and Expression 9 allows us to reach the conclusion that oracle

Recall that the oracle operator is applied to register

Any computational procedure can be described in production system theory by specifying an appropriate set of production rules that are responsible for performing an adequate state evolution. This set of production rules can be applied in conjunctions with a unitary operator

Traditionally, throughout a computation set

Any approach to a universal model of quantum computation needs to focus on two main issues, namely: (1) how to circumvent the halting problem and (2) how to handle computations that do not terminate without disturbing the result of the procedure. In the next sections we describe our general procedure. We choose to focus first on the second requirement given that it provides a basis for model development by establishing the parallels between

Universal computation must allow for the possibility of non-termination, a characteristic that is is achievable through the ability to calculate

In order to answer this question we will first start by establishing some parallels between these concepts. Namely, consider the

The fact that we are able to perform such mappings hints at the possibility of being able to develop our own quantum equivalent of the

This process is illustrated in

Line 8 is responsible for applying the oracle alongside an initial state and all possible sequences of productions. Recall that register

Due to the probabilistic nature of Grover's algorithm there is also the possibility that the measurement will return a non halting state, even though

Each iteration of

Since we employ Grover's algorithm we do not need to measure specifically the halting register. Instead it is possible to perform a measurement on the entire Hilbert space of the system in order to verify if a final state is obtained. This type of a control structure is responsible for guaranteeing the same type of partial behaviour that can be found on the classical

Quantum iterative deepening search may seem inefficient, because each time we apply

By employing our proposal we are able to develop a quantum computational model with an inherent speedup relatively to its classical counterparts. Notice that this speedup is only obtained when searching through a search space with a branching factor of at least

The approach proposed in this work allows for the possibility of non-termination, without inherently interfering with the results of the quantum computation. This hints at the possibility that our approach can be applied to coherently simulate classical universal models of computation such as the Turing machine. Specifically, we are interested in determining what would be needed for our model of an iterative quantum production system to simulate any classical Turing machine?

We will begin by presenting a set of mappings between Turing machine concepts and production system concepts in a manner analogous to the trivial mapping described in

The Turing machine model utilises a

The only remaining issue resides in defining a control strategy

In this work we presented an approach for an iterative quantum production system with a built-in speedup mechanism and capable of the partial behaviour characteristic of

Our model is able to operate independently of whether the computation terminates or not, a requirement associated with universal models of computation. As a result, it becomes possible for our model to exhibit partial behaviour that does not disturb the overall result of the underlying quantum computational process. This result is possible since: (1) Grover's algorithm effectively allows one to obtain halting states, if they exist, with high probability upon system observation; and (2) the overall complexity of this proposition remains the same of the quantum search algorithm. This procedure enables the development of verification-based universal quantum computational models, which are capable of coherently simulating classical models of universal computation such as the Turing machine.