No existence of a linear algorithm for the one-dimensional Fourier phase retrieval

Fourier phase retrieval, which aims to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we provide a theoretical understanding of algorithms for the one-dimensional Fourier phase retrieval problem. Specifically, we demonstrate that if an algorithm exists which can reconstruct an arbitrary signal $x\in C^N$ in $\text{Poly}\left(N\right)\log (1/ϵ)$ time to reach ϵ-precision from its magnitude of discrete Fourier transform and its initial value $x\left(0\right)$, then $P=\mathrm{NP}$. This partially elucidates the phenomenon that, despite the fact that almost all signals are uniquely determined by their Fourier magnitude and the absolute value of their initial value $\left|x\right(0\left)\right|$, no algorithm with theoretical guarantees has been proposed in the last few decades. Our proofs employ the result in computational complexity theory that the Product Partition problem is NP-complete in the strong sense.