One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the Deutsch-Jozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor's quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an "alphabetic building blocks approach," we use these primitives to form an "algorithmic toolkit" for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover's algorithm is most surprisingly "almost" a QHS algorithm, and how this result suggests the possibility of an even more complete "algorithmic tookit" beyond the QHS algorithms.One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the Deutsch-Jozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor's quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an "alphabetic building blocks approach," we use these primitives to form an "algorithmic toolkit" for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover's algorithm is most surprisingly "almost" a QHS algorithm, and how this result suggests the possibility of an even more complete "algorithmic tookit" beyond the QHS algorithms.

arXiv:quant-ph/0607046Jul 6, 20060 citations

Methods

Adam Iaizzi

Springer thesesJan 1, 20180 citations

Manifold Learning for Dimensionality Reduction: Quantum Isomap algorithm

Weijun Feng, Gongde Guo, Kai Yu +2 more

Isomap algorithm is a representative manifold learning algorithm. The algorithm simplifies the data analysis process and is widely used in neuroimaging, spectral analysis and other fields. However, the classic Isomap algorithm becomes unwieldy when dealing with large data sets. Our object is to accelerate the classical algorithm with quantum computing, and propose the quantum Isomap algorithm. The algorithm consists of two sub-algorithms. The first one is the quantum Floyd algorithm, which calculates the shortest distance for any two nodes. The other is quantum Isomap algorithm based on quantum Floyd algorithm, which finds a low-dimensional representation for the original high-dimensional data. Finally, we analyze that the quantum Floyd algorithm achieves exponential speedup without sampling. In addition, the time complexity of quantum Isomap algorithm is $O(dNpolylogN)$ . Both algorithms reduce the time complexity of classical algorithms.Isomap algorithm is a representative manifold learning algorithm. The algorithm simplifies the data analysis process and is widely used in neuroimaging, spectral analysis and other fields. However, the classic Isomap algorithm becomes unwieldy when dealing with large data sets. Our object is to accelerate the classical algorithm with quantum computing, and propose the quantum Isomap algorithm. The algorithm consists of two sub-algorithms. The first one is the quantum Floyd algorithm, which calculates the shortest distance for any two nodes. The other is quantum Isomap algorithm based on quantum Floyd algorithm, which finds a low-dimensional representation for the original high-dimensional data. Finally, we analyze that the quantum Floyd algorithm achieves exponential speedup without sampling. In addition, the time complexity of quantum Isomap algorithm is $O(dNpolylogN)$. Both algorithms reduce the time complexity of classical algorithms.

arXiv:2212.03599Dec 7, 20220 citations

Comparison of several decoding algorithms for LDPC

Pei Bao-chen

The performance and application future of the channel coding is determined by the decoding algorithms.The advantage of the LDPC is the decoding algorithm. This paper introduces several decoding algorithms for LDPC including Log-BP algorithm,Min-Sum algorithm and modified Min-Sum algorithm. Simulations are done for the decoding algorithms which is sent through additive white Gaussian noise (AWGN) channel. The analysis of the Bit Error Performance and the complexity for decoding algorithms is shown that Modified Min-Sum algorithm is the best algorithm which fits for the application of the hardware.The performance and application future of the channel coding is determined by the decoding algorithms.The advantage of the LDPC is the decoding algorithm. This paper introduces several decoding algorithms for LDPC including Log-BP algorithm,Min-Sum algorithm and modified Min-Sum algorithm. Simulations are done for the decoding algorithms which is sent through additive white Gaussian noise (AWGN) channel. The analysis of the Bit Error Performance and the complexity for decoding algorithms is shown that Modified Min-Sum algorithm is the best algorithm which fits for the application of the hardware.

Radio and communications technologyJan 1, 20060 citations

Optimization

Robert Wille, Rolf Drechsler

Springer eBooksJan 1, 20100 citations

O(dNpolylogN)

. Both algorithms reduce the time complexity of classical algorithms.