An Extended Row and Column Method for Solving Linear Systems on a Quantum Computer

To solve the linear systems of equations Ax = b on a quantum computer, Shao and Xiang proposed a quantum version of row and column methods by establishing unitary operators in each iteration step based on the block-encoding technique in Shao and Xiang (Phys. Rev. A 101, 022322, 2020]. In this paper, we generalize this strategy for solving the linear systems of equations Ax = b by an extended randomized row and column method on a quantum computer, where both the rows and columns of coefficient matrix are used via the block-encoding technique. If the quantum states are effectively prepared, compared with its traditional counterpart, our quantum iterative algorithm achieves an exponential speedup in the problem dimension n. The complexity of the quantum extended row and column method is $O({{\kappa _{s}^{2}}(A)}(\log n) \log {1}/{\epsilon })$ , where κs(A) is the scaled condition number of A, and 𝜖 is the error.