Quantum Algorithms

Samuel J. Lomonaco, Louis H. Kauffman

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.

Dipankar Bhattacharyya, Jyotirmoy Guha

Neha Junagade, Sheetal Umesh Bhandari, Rachana Y Patil

Quantum computing is a rapidly growing field. Quantum algorithms play a significant role and have applications in cryptography, machine learning, optimization, and computational issues [1][2]. They quantum features like entanglement and superposition that help carryout tasks that would be difficult for a classical computer. More Interest has been developed in Quantum algorithms due to their ability to solve specific computational problems quicker than that in classical algorithms [3]. These algorithms use fundamentals of quantum physics to carryout calculations quickly and effectively than traditional algorithms. Here in-depth analysis of quantum algorithms and classical algorithms is done by looking at the time and space complexity of quantum and classical algorithms including their capacity to tackle specific types of issues.First, we conduct a general comparison of Classical and Quantum Algorithm along with Background and motivation. We also discuss Simon's algorithm first and then Grover's algorithm in depth and their comparison with few other algorithms in a table format [5][6][7][8]. We also cover some recent advancements, Quantum supremacy, Post-Quantum Cryptography, 8 Quantum Artificial Intelligence, Hybrid Quantum-Classical Algorithms and few other important parameters. We also discuss the plausible impact of quantum algorithms on cybersecurity and cryptography [13][9]. Overall, this comparative study shows the uses and challenges involved in implementing quantum algorithms while offering an overview of quantum algorithms. We conclude by the fundamental ideas of quantum algorithms. Adding to this we also discuss the challenges and limitations of using quantum algorithms in practically. At the end, we provide a summary of the primary findings from this comparative study of Quantum and classical algorithms.

Venkateswaran Kasirajan

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.