We present a quantum algorithm for fitting a linear regression model to a given data set using the least-squares approach. Differently from previous algorithms which yield a quantum state encoding the optimal parameters, our algorithm outputs these numbers in the classical form. So by running it once, one completely determines the fitted model and then can use it to make predictions on new data at little cost. Moreover, our algorithm works in the standard oracle model, and can handle data sets with nonsparse design matrices. It runs in time $\text{poly}({log}_{2}(N),d,\ensuremath{\kappa},1/\ensuremath{\epsilon})$, where $N$ is the size of the data set, $d$ is the number of adjustable parameters, $\ensuremath{\kappa}$ is the condition number of the design matrix, and $\ensuremath{\epsilon}$ is the desired precision in the output. We also show that the polynomial dependence on $d$ and $\ensuremath{\kappa}$ is necessary. Thus, our algorithm cannot be significantly improved. Furthermore, we also give a quantum algorithm that estimates the quality of the least-squares fit (without computing its parameters explicitly). This algorithm runs faster than the one for finding this fit, and can be used to check whether the given data set qualifies for linear regression in the first place.We present a quantum algorithm for fitting a linear regression model to a given data set using the least-squares approach. Differently from previous algorithms which yield a quantum state encoding the optimal parameters, our algorithm outputs these numbers in the classical form. So by running it once, one completely determines the fitted model and then can use it to make predictions on new data at little cost. Moreover, our algorithm works in the standard oracle model, and can handle data sets with nonsparse design matrices. It runs in time $\text{poly}({log}_{2}(N),d,\ensuremath{\kappa},1/\ensuremath{\epsilon})$, where $N$ is the size of the data set, $d$ is the number of adjustable parameters, $\ensuremath{\kappa}$ is the condition number of the design matrix, and $\ensuremath{\epsilon}$ is the desired precision in the output. We also show that the polynomial dependence on $d$ and $\ensuremath{\kappa}$ is necessary. Thus, our algorithm cannot be significantly improved. Furthermore, we also give a quantum algorithm that estimates the quality of the least-squares fit (without computing its parameters explicitly). This algorithm runs faster than the one for finding this fit, and can be used to check whether the given data set qualifies for linear regression in the first place.

arXiv:1402.0660Physical Review AJul 31, 2017144 citations

Quantum regularized least squares solver with parameter estimate

Changpeng Shao, Hua Xiang

Quantum Information ProcessingFeb 14, 20207 citations

Quantum Algorithm for Data Fitting

Nathan Wiebe, Daniel Braun, Seth Lloyd

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. 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 are pure quantum states, 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.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. 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 are pure quantum states, 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.

arXiv:1204.5242Physical Review LettersAug 2, 2012528 citations

Quantum generalized least squares method in system identification

Sadegh Kalantari, Ali Madady, Mehdi Ramezani

Quantum Information ProcessingJun 12, 20242 citations

N

is the size of the data set,

d

is the number of adjustable parameters, $\ensuremath{\kappa}$ is the condition number of the design matrix, and $\ensuremath{\epsilon}$ is the desired precision in the output. We also show that the polynomial dependence on

d

and $\ensuremath{\kappa}$ is necessary. Thus, our algorithm cannot be significantly improved. Furthermore, we also give a quantum algorithm that estimates the quality of the least-squares fit (without computing its parameters explicitly). This algorithm runs faster than the one for finding this fit, and can be used to check whether the given data set qualifies for linear regression in the first place.