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

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.