A simple analysis of a quantum-inspired algorithm for solving low-rank linear systems

TL;DR

This paper presents a straightforward, elementary analysis of a quantum-inspired sampling algorithm for efficiently approximating solutions to low-rank linear systems, improving understanding of its complexity and capabilities.

Contribution

It introduces a simple, self-contained analysis of a quantum-inspired algorithm for solving low-rank linear systems, simplifying previous complex proofs.

Findings

Algorithm produces a compressed solution approximation with error less than ε.

Time complexity depends polynomially on condition numbers and inverse error.

Queries and sampling of the solution vector are efficient given access to specific samplers.

Abstract

We describe and analyze a simple algorithm for sampling from the solution $\mathbf{x}^* := \mathbf{A}^+\mathbf{b}$ to a linear system $\mathbf{A}\mathbf{x} = \mathbf{b}$. We assume access to a sampler which allows us to draw indices proportional to the squared row/column-norms of $\mathbf{A}$. Our algorithm produces a compressed representation of some vector $\mathbf{x}$ for which $\|\mathbf{x}^* - \mathbf{x}\| < \varepsilon \|\mathbf{x}^* \|$ in $\widetilde{O}(\kappa_{\mathsf{F}}^4 \kappa^2 / \varepsilon^2)$ time, where $\kappa_{\mathsf{F}} := \|\mathbf{A}\|_{\mathsf{F}}\|\mathbf{A}^{+}\|$ and $\kappa := \|\mathbf{A}\|\|\mathbf{A}^{+}\|$. The representation of $\mathbf{x}$ allows us to query entries of $\mathbf{x}$ in $\widetilde{O}(\kappa_{\mathsf{F}}^2)$ time and sample proportional to the square entries of $\mathbf{x}$ in $\widetilde{O}(\kappa_{\mathsf{F}}^4 \kappa^6)$ time, assuming access to a sampler which allows us to draw indices proportional to the squared entries of any given row of $\mathbf{A}$. Our analysis, which is elementary, non-asymptotic, and fully self-contained, simplifies and clarifies several past analyses from literature including [Gily\'en, Song, and Tang; 2022, 2023] and [Shao and Montanaro; 2022].