An efficient quantum algorithm for independent component analysis

Independent component analysis (ICA) is a fundamental data processing technique to decompose the captured signals into as independent as possible components. Computing the contrast function, which serves as a measure of independence of signals, is vital in the separation process using ICA. This paper presents a quantum ICA algorithm which focuses on computing a specified contrast function on a quantum computer. Using the quantum acceleration in matrix operations, we efficiently deal with Gram matrices and estimate the contrast function with the complexity of $O(\epsilon_1^{-2}\mbox{poly}\log(N/\epsilon_1))$. This estimation subprogram, combined with the classical optimization framework, enables our quantum ICA algorithm, which exponentially reduces the complexity dependence on the data scale compared with classical algorithms. The outperformance is further supported by numerical experiments, while a source separation of a transcriptomic dataset is shown as an example of application.