A Quantum Cellular Automata architecture with nearest neighbor interactions using one quantum gate type

📝 Abstract

We propose an architecture based on Quantum cellular Automata which allows the use of only one type of quantum gates per computational step in order to perform nearest neighbor interactions. The model is built in partial steps, each one of them analyzed using nearest neighbor interactions, starting with single qubit operations and continuing with two qubit ones. The effectiveness of the model is tested and valuated by developing a quantum circuit implementing the Quantum Fourier Transform. The important outcome of this validation was that the operations are performed in a local and controlled manner thus reducing the error rate of each computational step.

💡 Analysis

We propose an architecture based on Quantum cellular Automata which allows the use of only one type of quantum gates per computational step in order to perform nearest neighbor interactions. The model is built in partial steps, each one of them analyzed using nearest neighbor interactions, starting with single qubit operations and continuing with two qubit ones. The effectiveness of the model is tested and valuated by developing a quantum circuit implementing the Quantum Fourier Transform. The important outcome of this validation was that the operations are performed in a local and controlled manner thus reducing the error rate of each computational step.

📄 Content

A Quantum Cellular Automata architecture with nearest neighbor interactions using one quantum gate type. Dimitris Ntalaperas and Nikos Konofaos Department of Informatics Aristotle University of Thessaloniki, Biology Building, Main University Campus, 54124 Thessaloniki, Greece

Abstract We propose an architecture based on Quantum cellular Automata which allows the use of only one type of quantum gates per computational step in order to perform nearest neighbor interactions. The model is built in partial steps, each one of them analyzed using nearest neighbor interactions, starting with single qubit operations and continuing with two qubit ones. The effectiveness of the model is tested and valuated by developing a quantum circuit implementing the Quantum Fourier Transform. The important outcome of this validation was that the operations are performed in a local and controlled manner thus reducing the error rate of each computational step.

1 Introduction The notion of taking advantage of quantum mechanical principles in order to design computers that could, in principle, be more powerful than their classical counterparts was firstly proposed by Feynman [1], while Margolus [2] demonstrated the existence of computational models that were based on quantum mechanics and could perform reversible computation in the form of Reversible Cellular Automata (RCA). Various quantum computational models, being able to exhibit universal behavior, have also been defined and shown to be equivalent. In particular, Deutsch [3] defined the formulation of the Quantum Turing Machine (QTM), Yao [4] formulated the Quantum Circuit Model and demonstrated its equivalence with the QTM, while Watrous [5] defined one-dimensional Quantum Cellular Automata and showed that there is an efficient way to simulate a subclass of them by a QTM. Algorithms that harness the power of quantum computational model and outperform the corresponding classical ones have also been developed; Shor [6] has designed a quantum algorithm that can factorize composite numbers achieving a superpolynomial speedup, while Grover [7] demonstrated how searching in an unsorted database can be performed by a quantum algorithm with a quadratic speedup. Physical implementation of a quantum computer can pose a challenge, since quantum systems tend to decohere due to unwanted couplings with the environment. Various implementation architectures that try to overcome this problem have however been proposed [8] [9]. One common characteristics of the vast majority of implementation architectures is that they allow for only neighboring qubits to interact. Due to this limitation a number of schemes have been studied, in which a general quantum algorithm can be converted to an equivalent one, allowing only nearest neighbor interactions [10][11]. Typically, these

schemes convert a quantum circuit to one consisting of gates acting only on neighboring qubits at the cost of introducing a number of additional gates to the original circuit.
In this paper, an architecture that enforces only nearest neighbor interactions and the application of only one type of a quantum gate per computational step is introduced. This new architecture is based on Quantum Cellular Automata (QCA). The goals of the new architecture are: a) To convert a generic quantum circuit to an equivalent one having only nearest neighbor interactions and b) to allow only a specific quantum gate to act on all the qubits for the duration of one computational step. This latter limitation is introduced since in many implementations it is difficult to localize signals corresponding to two different quantum gates. In the architecture presented, a single signal is applied in each step and the same error is introduced to all qubit states. The only non local interactions allowed are those of quantum teleportation which have been shown to have a nearly zero error rate in various implementation schemes [12]. The present paper is structured as follows: First, an overview of a QCA is given along with an overview of specific type of QCA proposed by Karafyllidis [13] , which operate by applying a single two-qubit quantum gate over the whole quantum register in each step. Afterwards, an overview of nearest neighbor architectures is presented and it is demonstrated how quantum gates involving qubits over an arbitrary distance can be performed by using only local interactions and quantum teleportation in a 2D Grid model architecture which is based in ideas introduced by Rosenbaum[14] Then, we introduce our proposal where the two techniques are combined to this new architecture that retains the advantages of both methods. The execution of an arbitrary quantum gate is demonstrated in this new model and the amount of extra operations required per gate is calculated. Finally, an example of developing a quantum circuit implementing the Quantum Fourier Trans

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