Quantum Computing Techniques for Numerical Linear Algebra in Computational Mathematics

Quantum computing is a new and exciting area of computational mathematics that has the ability to solve very hard problems that traditional computing methods have not been able to solve for a long time. This abstract goes into detail about how quantum computing can be used in numerical linear algebra, which is an important part of computational mathematics that is used in many fields, such as science, engineering, and data analysis. The idea behind quantum computing is new. It uses quantum bits (qubits) and quantum gates to do calculations in a very different way, based on the rules of quantum physics. Because of this change in thinking, quantum computers might be able to solve some numerical linear algebra problems a lot faster than traditional computers. The outline then talks about some of the most important quantum computing methods and algorithms that are made for numerical linear algebra problems. There are quantum versions of standard algorithms like matrix multiplication, Gaussian elimination, and singular value decomposition. There are also new algorithms that are intended to use quantum parallelism and interference to speed up computations. For situations where full-scale quantum gear is not yet available, quantum-inspired methods are also talked about. These blend conventional and quantum techniques. The abstract talks about the difficulties and chances of using quantum algorithms for numerical linear algebra on quantum hardware systems that are available now and in the near future. It talks about things like error correction, qubit coherence times, and making efficient quantum circuit implementations. Lastly, the abstract talks about how quantum computing might change computational mathematics. It imagines that advances in quantum computing could help solve numerical linear algebra problems that were previously impossible to solve. This could have huge effects on many fields, from machine learning and optimization to quantum chemistry and cryptography. In the end, it stresses how important it is for mathematicians, computer scientists, physicists, and engineers to work together across disciplines in order to fully utilize the transformative power of quantum computing in numerical linear algebra and other areas.