Quantum Data Fitting

TL;DR

This paper introduces a quantum algorithm that efficiently performs least-squares fitting on large datasets, approximates functions, and estimates quantum states, offering advantages over classical methods and quantum tomography.

Contribution

It presents a novel quantum algorithm for data fitting, state estimation, and function approximation, extending the capabilities of quantum linear systems algorithms.

Findings

Efficient least-squares fitting on exponentially large datasets

Ability to find concise data approximations with bounded error

Application as an alternative to quantum state tomography

Abstract

We provide a new quantum algorithm that efficiently determines the quality of a least-squares fit over an exponentially large data set by building upon an algorithm for solving systems of linear equations efficiently (Harrow et al., Phys. Rev. Lett. {\bf 103}, 150502 (2009)). In many cases, our algorithm can also efficiently find a concise function that approximates the data to be fitted and bound the approximation error. In cases where the input data is a pure quantum state, the algorithm can be used to provide an efficient parametric estimation of the quantum state and therefore can be applied as an alternative to full quantum state tomography given a fault tolerant quantum computer.