Quantum Linear System Algorithm for Dense Matrices

Solving linear systems of equations is a frequently encountered problem in machine learning and optimization. Given a matrix A and a vector b the task is to find the vector x such that Ax=b. We describe a quantum algorithm that achieves a sparsity-independent runtime scaling of O(κ²√npolylog(n)/ε) for an n×n dimensional A with bounded spectral norm, where κ denotes the condition number of A, and ε is the desired precision parameter. This amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices, and poses a new state of the art for solving dense linear systems on a quantum computer. Furthermore, an exponential improvement is achievable if the rank of A is polylogarithmic in the matrix dimension. Our algorithm is built upon a singular value estimation subroutine, which makes use of a memory architecture that allows for efficient preparation of quantum states that correspond to the rows of A and the vector of Euclidean norms of the rows of A.