  The Fourier Transform is one of the most important linear transformations
used in science and engineering. Cooley and Tukey's Fast Fourier Transform
(FFT) from 1964 is a method for computing this transformation in time $O(n\log
n)$. Achieving a matching lower bound in a reasonable computational model is
one of the most important open problems in theoretical computer science.
  In 2014, improving on his previous work, Ailon showed that if an algorithm
speeds up the FFT by a factor of $b=b(n)\geq 1$, then it must rely on
computing, as an intermediate "bottleneck" step, a linear mapping of the input
with condition number $\Omega(b(n))$. Our main result shows that a factor $b$
speedup implies existence of not just one but $\Omega(n)$ $b$-ill conditioned
bottlenecks occurring at $\Omega(n)$ different steps, each causing information
from independent (orthogonal) components of the input to either overflow or
underflow. This provides further evidence that beating FFT is hard. Our result
also gives the first quantitative tradeoff between computation speed and
information loss in Fourier computation on fixed word size architectures. The
main technical result is an entropy analysis of the Fourier transform under
transformations of low trace, which is interesting in its own right.
