A mixed-precision quantum-classical algorithm for solving linear systems

We address the problem of solving a system of linear equations via the Quantum Singular Value Transformation (QSVT). One drawback of the QSVT algorithm is that it requires huge quantum resources if we want to achieve an acceptable accuracy. To reduce the quantum cost, we propose a hybrid quantum-classical algorithm that improves the accuracy and reduces the cost of the QSVT by adding iterative refinement in mixed-precision A first quantum solution is computed using the QSVT, in low precision, and then refined in higher precision until we get a satisfactory accuracy. For this solver, we present an error and complexity analysis, and first experiments using the quantum software stack myQLM.