In this paper, we analyze the effectiveness of quantum algorithms to solve some classical computing complexities. In fact, we focus in this study on several famous quantum algorithms, where we discussed their impact on classical computing using in computer science.
Quantum computing is necessary to be described as a powerful tool is the new generation of computational power that works based on the principles of quantum mechanics, allowing the solving of problems that are intractable for classical computers. Due to the utilization of such phenomena as superposition, quantum entanglement and quantum interference, quantum computers are said to be able to solve problems in such areas as cryptography, artificial intelligence, optimization problems and material sciences, among others. This paper attempts to examine the basic concepts of quantum computing while focusing on their different aspects and their potential impacts. This study its experience and potential in its current form and in future applications through examples, illustrations, and theoretical discussion. Specific problems, including error correction, hardware scalability, and requirements for resources to address, make the paper a comprehensive overview of the quantum computing potential and weaknesses. Moving towards practical examples, the discussion covers all aspects of the book, providing readers with a holistic understanding of how quantum computing is going to change technological paradigms in the forthcoming decades.
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.
The anticipated applications of quantum computers span across science and industry, ranging from quantum chemistry and many-body physics to optimization, finance, and machine learning. Proposed quantum solutions in these areas typically combine multiple quantum algorithmic primitives into an overall quantum algorithm, which must then incorporate the methods of quantum error correction and fault tolerance to be implemented correctly on quantum hardware. As such, it can be difficult to assess how much a particular application benefits from quantum computing, as the various approaches are often sensitive to intricate technical details about the underlying primitives and their complexities. Here we present a survey of several potential application areas of quantum algorithms and their underlying algorithmic primitives, carefully considering technical caveats and subtleties. We outline the challenges and opportunities in each area in an "end-to-end" fashion by clearly defining the problem being solved alongside the input-output model, instantiating all "oracles," and spelling out all hidden costs. We also compare quantum solutions against state-of-the-art classical methods and complexity-theoretic limitations to evaluate possible quantum speedups. The survey is written in a modular, wiki-like fashion to facilitate navigation of the content. Each primitive and application area is discussed in a standalone section, with its own bibliography of references and embedded hyperlinks that direct to other relevant sections. This structure mirrors that of complex quantum algorithms that involve several layers of abstraction, and it enables rapid evaluation of how end-to-end complexities are impacted when subroutines are altered.
Abstract Solving linear systems of equations is of vital importance in various fields of science. With the scale of calculation soaring, classical computers are intractable, and the advantage of quantum computing is reflected. Quantum algorithm has exponential acceleration in solving the linear systems A x ⃗ = b ⃗ for sparse and well-conditioned coefficient matrix A with dimension N × N . In this paper, we present a quantum algorithm to solve matrix equations of the form AX = B , where X = ( x ⃗ 1 , x ⃗ 2 , … , x ⃗ n ) , B = ( b ⃗ 1 , b ⃗ 2 , … , b ⃗ n ) , x ⃗ k and b ⃗ k are unit vectors. The problem can be intuitively treated as solving n linear systems simultaneously, which will needs O ( n log N ) complexity by the famous HHL algorithm. We demonstrate that in our algorithm the scale of complexity is O ( log n N ) . In the error analysis part, phase estimation is discussed in details.
Quantum computing is coming up as futuristic technology in engineering and science.The famous Sturm-Liouville (SL) differential equation along with some specific boundary conditions appear in many branches of physics as well as in other fundamental fields.The numerical solution of such boundary value problems (like heat transfer, potential problem etc.) is frequently computationally expensive, and traditional computers may not be able to deliver accurate solutions in an acceptable length of time.We propose quantum computing approach to solve such boundary value problems that will be more efficient and time solvent.The present research demands that the Harrow-Hassidim-Lloyd (HHL) quantum algorithm provides an efficient technique for solving such boundary value problems that arise in many branches of physical science and engineering.The prime focus of this research is to explore HHL quantum algorithm in detail.The quantum phase estimation algorithm is applied for solving eigen values and eigenvectors associated with the SL type of boundary value problem.As a case study, in this research, we have solved the steady state heat transfer problem with boundary conditions.Our computation is based on Qiskit module made by IBM and available as a library in Python.A detailed analysis of the computational complexity of our algorithm that compares its performance with classical methods on a range of boundary value problems has been discussed.Role of Tridiagonal Toeplitz matrix is revealed.Results demonstrate that the method of quantum algorithm for solving boundary value problems can significantly outperform classical methods, especially for larger problem sizes.This work represents a significant methodology in solving important equations of physics and mathematics in future quantum computers.