Hybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines

We propose a realistic hybrid classical-quantum linear solver to solve systems of linear equations of a specific type, and demonstrate its feasibility using Qiskit on IBM Q systems. This algorithm makes use of quantum random walk that runs in $mathcal{O}(Nlog(N))$ time on a quantum circuit made of $mathcal{O}(log(N))$ qubits. The input and output are classical data, and so can be easily accessed. It is robust against noise, and ready for implementation in applications such as machine learning.