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IEEE International Conference on Power, Electronics and Computer Applications (ICPECA)

Quantum Coins for Non-Power-of-Two States - Implementation and Performance on Quantum Walk and Quantum Walk Search Algorithms

Tse Kai Chan

Published Jan 21, 2022• 0 citations
Publisher

Abstract

Quantum coin is an essential element of the discrete-time quantum walk, the quantum analog of random walk. Most existing papers relating to quantum coin discussed the implementation and performance of the coin on power-of-two states and degrees of graphs. In this paper, I proposed and analyzed three alternative coins–binary state coin, W state coin, and Walsh-Hadamard coin–that possessed the property to perform on all states and degrees of graphs. I discussed the implementation and the performance of the three coins in the application of Quantum Walk and Quantum Walk Search. I compared and discussed the advantages, disadvantages, and errors of the three coins on a quantum simulator and a NISQ computer (i.e. IBM Q). In the end, I presented further applications for the three coins in quantum-walk-based algorithms and other potential topics in quantum computing.

DOI
10.1109/icpeca53709.2022.9719105

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Quantum coin is an essential element of the discrete-time quantum walk, the quantum analog of random walk. Most existing papers relating to quantum coin discussed the implementation and performance of the coin on power-of-two states and degrees of graphs. In this paper, I proposed and analyzed three alternative coins–binary state coin, W state coin, and Walsh-Hadamard coin–that possessed the property to perform on all states and degrees of graphs. I discussed the implementation and the performance of the three coins in the application of Quantum Walk and Quantum Walk Search. I compared and discussed the advantages, disadvantages, and errors of the three coins on a quantum simulator and a NISQ computer (i.e. IBM Q). In the end, I presented further applications for the three coins in quantum-walk-based algorithms and other potential topics in quantum computing.
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In this thesis we have focused on two topics: Discrete Quantum Walks and Quantum Image Processing. Our work is a contribution within the field of quantum computation from the perspective of a computer scientist. With the purpose of finding new techniques to develop quantum algorithms, there has been an increasing interest in studying Quantum Walks, the quantum counterparts of classical random walks. Our work in quantum walks begins with a critical and comprehensive assessment of those elements of classical random walks and discrete quantum walks on undirected graphs relevant to algorithm development. We propose a model of discrete quantum walks on an infinite line using pairs of quantum coins under different degrees of entanglement, as well as quantum walkers in different initial state configurations, including superpositions of corresponding basis states. We have found that the probability distributions of such quantum walks have particular forms which are different from the probability distributions of classical random walks. Also, our numerical results show that the symmetry properties of quantum walks with entangled coins have a non-trivial relationship with corresponding initial states and evolution operators. In addition, we have studied the properties of the entanglement generated between walkers, in a family of discrete Hadamard quantum walks on an infinite line with one coin and two walkers. We have found that there is indeed a relation between the amount of entanglement available in each step of the quantum walk and the symmetry of the initial coin state. However, as we show with our numerical simulations, such a relation is not straightforward and, in fact, it can be counterintuitive. Quantum Image Processing is a blend of two fields: quantum computation and image processing. Our aim has been to promote cross-fertilisation and to explore how ideas from quantum computation could be used to develop image processing algorithms. Firstly, we propose methods for storing and retrieving images using non-entangled and entangled qubits. Secondly, we study a case in which 4 different values are randomly stored in a single qubit, and show that quantum mechanical properties can, in certain cases, allow better reproduction of original stored values compared with classical methods. Finally, we briefly note that entanglement may be used as a computational resource to perform hardware-based pattern recognition of geometrical shapes that would otherwise require classical hardware and software.
Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator, constant in both space and time, and a localized initial state using numerical modeling. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study on general coin operators also includes the popular coins – Hadamard, Grover, and Fourier.
Quantum walk (QW) is the quantum analog of the random walk. QW is an integral part of the development of numerous quantum algorithms. Hence, an in-depth understanding of QW helps us to grasp the quantum algorithms. We revisit the one-dimensional discrete-time QW and discuss basic steps in detail by incorporating the most general coin operator. We investigate the impact of each parameter of the general coin operator on the probability distribution of the quantum walker. We show that by tuning the parameters of the general coin, one can regulate the probability distribution of the walker. We provide an algorithm for the one-dimensional quantum walk driven by the general coin operator. The study conducted on general coin operator also includes the popular coins -- Hadamard, Grover, and Fourier coins.
Quantum walks are a powerful tool for developing efficient algorithms in quantum computing. This research explores two discrete-time one-dimensional quantum walks where the coin operator varies along even and odd positions on the line. We find closed-form expressions for the coefficients of the wave function for both walks and also arrive at a formula for the probability distribution for one of the walks. A significant discovery is a way to model the well-known Hadamard walk using two alternating coins.