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The Deutsch-Josza algorithm for n-qudits

Gabriela Mogos

Published Nov 21, 2007• 2 citations
OpenAlex

Abstract

Abstract: Over the last decade has been a lot of interest in new types of computable machines, due partly over the fears on conventional, classical, computing. Research by leading physicists, mathematicians and computer scientists has shown quantum computers could be the future, as they may be able to solve some computationally difficult problems much faster than is currently possible on a classical computer. David Deutsch has already created a working model for a quantum computer, the quantum Turing machine or QTM Deutsch [1985]. This has been proved as a model for computation on a quantum and has allowed algorithms to be developed. Deutsch-Josza’s algorithm, as all known quantum algorithms that provide exponential speedup over classical systems do, answers a question about a global property of a solution space. This paper describes the generalization of the Deutsch-Josza algorithm to n d-dimensional quantum systems or qudits. Key–Words: quantum computing, quantum dits, quantum algorithm 1

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Abstract: Over the last decade has been a lot of interest in new types of computable machines, due partly over the fears on conventional, classical, computing. Research by leading physicists, mathematicians and computer scientists has shown quantum computers could be the future, as they may be able to solve some computationally difficult problems much faster than is currently possible on a classical computer. David Deutsch has already created a working model for a quantum computer, the quantum Turing machine or QTM Deutsch [1985]. This has been proved as a model for computation on a quantum and has allowed algorithms to be developed. Deutsch-Josza’s algorithm, as all known quantum algorithms that provide exponential speedup over classical systems do, answers a question about a global property of a solution space. This paper describes the generalization of the Deutsch-Josza algorithm to n d-dimensional quantum systems or qudits. Key–Words: quantum computing, quantum dits, quantum algorithm 1
In 1985, David Deutsch challenged the Church-Turing thesis by stating that his quantum model of computation "could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine". While this is thought to be true in general, there is usually no way of knowing that the corresponding classical algorithms are the best possible solutions. Here we provide an efficient classical simulation of the Deutsch-Jozsa algorithm, which was one of the first examples of quantum computational speed-up. Our conclusion is that the Deutsch-Jozsa quantum algorithm owes its speed-up to resources that are not necessarily quantum-mechanical, and when compared with the classical simulation offers no speed-up at all.
We present the new model of paraconsistent Turing machines and revise the models of quantum Turing machines and quantum circuits, stressing the concepts of entangled states and quantum parallelism which are important features for efficient quantum algorithms. We revise a quantum circuit that solves the so-called Deutsch’s problem, and then provide another solution to Deutsch’s problem by means of paraconsistent Turing machines. This raises some stimulating questions, and we close the paper by discussing some relationships between paraconsistent Turing machines and quantum Turing machines.
Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N – 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithm’s known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.
In this article, we present an introduction to quantum computing (QC) tailored for computing professionals such as programmers, machine learning engineers, and data scientists. Our approach abstracts away the physics underlying QC, which can be challenging to grasp, and frames it as a model of computation similar to, for instance, Turing machines. This helps readers grasp the fundamental principles of QC from a purely logical perspective. We begin by defining quantum states and qubits, establishing their mathematical representation and role in computation. We introduce fundamental concepts such as basis states, quantum gates, and tensor products, illustrating how these form the building blocks of quantum computation. Then we present the Deutsch-Josza algorithm, one of the simplest quantum algorithms that demonstrate how quantum computers can outperform classical computers. Finally, we provide guidance for further study, recommending resources for those interested in exploring quantum algorithms, implementations, and industry applications.
We want in this article to show the usefulness of Quantum Turing Machine (QTM) in a high-level didactic context as well as in theoretical studies. We use QTM to show its equivalence with quantum circuit model for Deutsch and Deutsch-Jozsa algorithms. Further we introduce a strategy of translation from Quantum Circuit to Quantum Turing models by these examples. Moreover we illustrate some features of Quantum Computing such as superposition from a QTM point of view and starting with few simple examples very known in Quantum Circuit form.
Deutsch's problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutsch's quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various de-quantised solutions for Deutsch's quantum algorithm—classical solutions which are as efficient as the quantum one—have been proposed in the literature. These solutions are based on the possibility of classically simulating "superpositions", a key ingredient of quantum algorithms, in particular, Deutsch's algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous de-quantisations of Deutsch's quantum algorithm, the resulting de-quantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their "observational power".