Solving linear systems of equations is a frequently encountered problem in machine learning and optimization. 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 $\mathcal{O}\mathbf{(}{\kappa }^2\sqrt{n}\text{polylog}(n)/\epsilon \mathbf{)}$ for an $n\times n$ dimensional $A$ with bounded spectral norm, where $\kappa$ denotes the condition number of $A$, and $\epsilon$ 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 $A$.

• Received 25 May 2017
• Revised 10 November 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.050502