A Lower Bound for Fourier Transform Computation in a Linear Model Over 2x2 Unitary Gates Using Matrix Entropy

TL;DR

This paper establishes an $oldsymbol{ ext{Ω}(n ext{log} n)}$ lower bound for Fourier transform computation in a layered linear circuit model restricted to unitary 2x2 gates, using matrix entropy as a potential function.

Contribution

It introduces a new lower bound proof for Fourier transform computation in a specific linear model, replacing determinant-based methods with matrix entropy.

Findings

Lower bound of Ω(n log n) for Fourier transform in the model

FFT algorithm fits within the restricted model

Matrix entropy is a promising potential function for lower bounds

Abstract

Obtaining a non-trivial (super-linear) lower bound for computation of the Fourier transform in the linear circuit model has been a long standing open problem. All lower bounds so far have made strong restrictions on the computational model. One of the most well known results, by Morgenstern from 1973, provides an $\Omega(n \log n)$ lower bound for the \emph{unnormalized} FFT when the constants used in the computation are bounded. The proof uses a potential function related to a determinant. The determinant of the unnormalized Fourier transform is $n^{n/2}$, and thus by showing that it can grow by at most a constant factor after each step yields the result. This classic result, however, does not explain why the \emph{normalized} Fourier transform, which has a unit determinant, should take $\Omega(n\log n)$ steps to compute. In this work we show that in a layered linear circuit model restricted to unitary $2\times 2$ gates, one obtains an $\Omega(n\log n)$ lower bound. The well known FFT works in this model. The main argument concluded from this work is that a potential function that might eventually help proving the $\Omega(n\log n)$ conjectured lower bound for computation of Fourier transform is not related to matrix determinant, but rather to a notion of matrix entropy.