Quantum multirow iteration algorithm for linear systems with nonsquare coefficient matrices

In the field of quantum linear system algorithms, quantum computing has realized exponential computational advantages over classical computing. However, the focus has been on square coefficient matrices, with few quantum algorithms addressing nonsquare matrices. Towards this kind of problems defined by Ax = b where A E Rm`n, we propose a quantum algorithm inspired by the classical multirow iteration method and provide an explicit quantum circuit based on the quantum comparator and quantum random access memory. The time complexity of our quantum multirow iteration algorithm is O(K log2 m), with K representing the number of iteration steps, which demonstrates an exponential speedup compared to the classical version.. Based on the convergence of the classical multirow iteration algorithm, we prove that our quantum algorithm converges faster than the quantum one-row iteration algorithm presented by Shao and Xiang [C. Shao and H. Xiang, Phys. Rev. A 101, 022322 (2020)]. Moreover, our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems and quadratic optimization problems.