Cai et al. reported an experimental demonstration of the simplest meaningful instance of this algorithm, that is, solving linear equations for various input vectors. The quantum circuit is optimized and compiled into a linear optical network with four photonic quantum bits (qubits) and four controlled logic gates, which is used to coherently implement every subroutine for this algorithm. For various input vectors, the quantum computer gives solutions for the linear equations with reasonably high precision, ranging from fidelities of 0.825 to 0.993. Barz et al. On February 5, 2013, Stefanie Barz and co-workers demonstrated the quantum algorithm for linear systems of equations on a photonic quantum computing architecture. This implementation used two consecutive entangling gates on the same pair of polarization-encoded qubits. Two separately controlled NOT gates were realized where the successful operation of the first was heralded by a measurement of two ancillary photons. Barz et al. found that the fidelity in the obtained output state ranged from 64.7% to 98.1% due to the influence of higher-order emissions from spontaneous parametric down-conversion. Pan et al. On February 8, 2013, Pan et al. reported a proof-of-concept experimental demonstration of the quantum algorithm using a 4-qubit nuclear magnetic resonance quantum information processor. The implementation was tested using simple linear systems of only 2 variables. Across three experiments they obtain the solution vector with over 96% fidelity. Wen et al. Another experimental demonstration using NMR for solving an 8*8 system was reported by Wen et al. in 2018 using the algorithm developed by Subaşı et al. Applications Quantum computers are devices that harness quantum mechanics to perform computations in ways that classical computers cannot. For certain problems, quantum algorithms supply exponential speedups over their classical counterparts, the most famous example being Shor's factoring algorithm. Few such exponential speedups are known, and those that are (such as the use of quantum computers to simulate other quantum systems) have so far found limited use outside the domain of quantum mechanics. This algorithm provides an exponentially faster method of estimating features of the solution of a set of linear equations, which is a problem ubiquitous in science and engineering, both on its own and as a subroutine in more complex problems. Electromagnetic scattering Clader et al. provided a preconditioned version of the linear systems algorithm that provided two advances. First, they demonstrated how a preconditioner could be included within the quantum algorithm. This expands the