A quantum linear system algorithm for dense matrices

Abstract

Solving linear systems of equations is a frequently encountered problem in machine learning and optimisation. Given a matrix $A$ and a vector $\mathbf{b}$ the task is to find the vector $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{b}$. We describe a quantum algorithm that achieves a sparsity-independent runtime scaling of $\mathcal{O}({\kappa }^2\Vert A{\Vert }_F\text{polylog}(n)/ϵ)$, where $n\times n$ is the dimensionality of $A$ with Frobenius norm $\Vert A{\Vert }_F$, $\kappa$ denotes the condition number of $A$, and $ϵ$ is the desired precision parameter. When applied to a dense matrix with spectral norm bounded by a constant, the runtime of the proposed algorithm is bounded by $\mathcal{O}({\kappa }^2\sqrt{n}\text{polylog}(n)/ϵ)$, which is a quadratic improvement over known quantum linear system algorithms. 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 and row Frobenius norms of $A$.