COMPARATIVE ANALYSIS OF CLASSICAL AND QUANTUM ALGORITHMS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS: THEORETICAL INSIGHTS AND BENCHMARKING CHALLENGES

Modern computers utilize a model based on a simple Turing machine concept. This study contains an extensive comparative review of classical and quantum algorithm approaches to solving a system of linear equations. The study details standard classical approaches like Gaussian Elimination or Conjugate Gradient Method and compares these with the algorithm of quantum theory algorithm HHL (Harrow-Hassidim-Lloyd). It considers the logic behind each of them, their benchmarks, scalability, implementation difficulties and practical use. Benchmarking results illustrate the fact that a classical approach continues to do especially good for small to moderate sized problems. However, quantum computation has the potential for an exponential speedup using algorithms like HHL in well-conditioned sparse matrices. The current limits of the hardware are discussed and future research directions needed to be able to utilize quantum computing for larger scale linear algebra problems are presented.