A quantum linear system algorithm for dense matrices - Wossnig, Leonard; Zhao, Zhikuan; Prakash, Anupam (arXiv.org, 2017)
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 \(\mathcal{O}(\kappa^2 \|A\|_F \text{polylog}(n)/\epsilon)\), where \(n\times n\) is the dimensionality of \(A\) with Frobenius norm \(\|A\|_F\), \(\kappa\) denotes the condition number of \(A\), and \(\epsilon\) 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 \(\mathcal{O}(\kappa^2\sqrt{n} \text{polylog}(n)/\epsilon)\), 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 \(A\).