# Quantum solvability of noisy linear problems by divide-and-conquer strategy

## TL;DR

This paper introduces a quantum divide-and-conquer algorithm for noisy linear problems that significantly reduces computational complexity and quantum sample size, enabling more efficient quantum solutions under certain noise conditions.

## Contribution

It proposes a novel quantum divide-and-conquer approach that decreases quantum sample size and complexity for noisy linear problems, outperforming existing methods.

## Key findings

- Achieves polynomial quantum-sample and time complexities.
- Reduces quantum sample size from exponential to sub-exponential.
- Identifies noise conditions under which quantum advantage is possible.

## Abstract

Noisy linear problems have been studied in various science and engineering disciplines. A class of "hard" noisy linear problems can be formulated as follows: Given a matrix $\hat{A}$ and a vector $\mathbf{b}$ constructed using a finite set of samples, a hidden vector or structure involved in $\mathbf{b}$ is obtained by solving a noise-corrupted linear equation $\hat{A}\mathbf{x} \approx \mathbf{b} + \boldsymbol\eta$, where $\boldsymbol\eta$ is a noise vector that cannot be identified. For solving such a noisy linear problem, we consider a quantum algorithm based on a divide-and-conquer strategy, wherein a large core process is divided into smaller subprocesses. The algorithm appropriately reduces both the computational complexities and size of a quantum sample. More specifically, if a quantum computer can access a particular reduced form of the quantum samples, polynomial quantum-sample and time complexities are achieved in the main computation. The size of a quantum sample and its executing system can be reduced, e.g., from exponential to sub-exponential with respect to the problem length, which is better than other results we are aware. We analyse the noise model conditions for such a quantum advantage, and show when the divide-and-conquer strategy can be beneficial for quantum noisy linear problems.