# A quantum linear system algorithm for dense matrices

## TL;DR

This paper introduces a quantum algorithm for solving dense linear systems with a runtime that improves quadratically over previous quantum methods, especially for matrices with bounded spectral norm, leveraging singular value estimation and efficient state preparation.

## Contribution

The paper presents a novel quantum linear system algorithm with sparsity-independent runtime scaling for dense matrices, improving efficiency over existing quantum algorithms.

## Key findings

- Achieves $ ilde{O}( au ext{polylog}(n)/ extepsilon)$ runtime for dense matrices.
- Provides a quadratic speedup over previous quantum algorithms for dense matrices.
- Utilizes a memory architecture for efficient quantum state preparation.

## 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 $\mathcal{O}(\kappa^2 \|A\|_F \text{polylog}(n)/\epsilon)$, 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 $A$.