Line spectra is an important phenomenon in many signal processing application including radar, sonar, communications, and spectroscopy. The estimation of a frequency component in a noisy sinusoidal signal for source localization and Doppler estimation is often required for practical systems such as cooperative communication, and vehicular communications. Here, we propose a quantum framework for the line spectrum estimation in the presence of observation noise and quantum error. We propose quantum algorithm, namely quantum line spectrum estimation (QLSE) to estimate the frequency of a sinusoidal signal from a quantum measurement probability histogram. Considering complex sinusoidal waveform, we show how to estimate single tone and multiple tone signals from the quantum circuit. We demonstrate a fully quantum computational model implemented on an IBM quantum machine and discuss quantum complexity advantage and error performance of the proposed algorithm in comparison with classical method. The experimental study shows promising result for super-resolution frequency estimation application with the proposed QLSE algorithm. The numerical results are given to support the proposition in the quantum regime. This manuscript is submitted to a journal with adequate modifications. The copyright may be transferred without notice, and this preprint may not be available after publication.