Abstract
We develop an efficient quantum implementation of an important signal processing algorithm for line spectral estimation: the matrix pencil method, which determines the frequencies and damping factors of signals consisting of finite sums of exponentially damped sinusoids. Our algorithm provides a quantum speedup in a natural regime where the sampling rate is much higher than the number of sinusoid components. Along the way, we develop techniques that are expected to be useful for other quantum algorithms as well—consecutive phase estimations to efficiently make products of asymmetric low rank matrices classically accessible and an alternative method to efficiently exponentiate non-Hermitian matrices. Our algorithm features an efficient quantum–classical division of labor: the time-critical steps are implemented in quantum superposition, while an interjacent step, requiring much fewer parameters, can operate classically. We show that frequencies and damping factors can be obtained in time logarithmic in the number of sampling points, exponentially faster than known classical algorithms.
Highlights
Algorithms for the spectral estimation of signals consisting of finite sums of exponentially damped sinusoids have a vast number of practical applications in signal processing
These range from imaging and microscopy [1], radar target identification [2], nuclear magnetic resonance spectroscopy [3], estimation of ultra wide-band channels [4], quantum field tomography [5, 6], power electronics [7], up to the simulation of atomic systems [8]
There are various so-called high resolution spectral estimation techniques that provide precisely such methods: they include matrix pencil methods (MPM) [9], Prony’s method [10], MUSIC [11], ESPRIT [12], and atomic norm denoising [13]. These techniques are superior to discrete Fourier transform (DFT) in instances with damped signals and close frequencies or small observation time T > 0 [14,15,16] and are preferred over of the Fourier transform in those applications laid out in [1,2,3,4,5, 7, 8]: the DFT resolution in the frequency domain Dw is proportional to 1 T, which is especially critical for poles that are close to each other
Summary
Any further distribution of factors of signals consisting of finite sums of exponentially damped sinusoids. Quantum algorithms as well—consecutive phase estimations to efficiently make products of asymmetric low rank matrices classically accessible and an alternative method to efficiently exponentiate non-Hermitian matrices. Our algorithm features an efficient quantum–classical division of labor: the time-critical steps are implemented in quantum superposition, while an interjacent step, requiring much fewer parameters, can operate classically. We show that frequencies and damping factors can be obtained in time logarithmic in the number of sampling points, exponentially faster than known classical algorithms
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