Linear Regression by Quantum Amplitude Estimation and its Extension to Convex Optimization

TL;DR

This paper introduces a quantum algorithm for linear regression with improved error dependence, achieving $O(psilon^{-1})$ complexity, and extends it to certain convex optimization problems, outperforming previous quantum methods.

Contribution

The authors develop a quantum linear regression algorithm with better error scaling and extend it to convex optimization, surpassing existing quantum approaches.

Findings

Quantum linear regression achieves $O(psilon^{-1})$ complexity.

Method maintains logarithmic dependence on data size $N_D$.

Extension to convex optimization broadens applicability.

Abstract

Linear regression is a basic and widely-used methodology in data analysis. It is known that some quantum algorithms efficiently perform least squares linear regression of an exponentially large data set. However, if we obtain values of the regression coefficients as classical data, the complexity of the existing quantum algorithms can be larger than the classical method. This is because it depends strongly on the tolerance error $\epsilon$: the best one among the existing proposals is $O(\epsilon^{-2})$. In this paper, we propose the new quantum algorithm for linear regression, which has the complexity of $O(\epsilon^{-1})$ and keeps the logarithmic dependence on the number of data points $N_D$. In this method, we overcome bottleneck parts in the calculation, which take the form of the sum over data points and therefore have the complexity proportional to $N_D$, using quantum amplitude estimation, and other parts classically. Additionally, we generalize our method to some class of convex optimization problems.