An n\log n Lower Bound for Fourier Transform Computation in the Well Conditioned Model

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 for over 40 years. An early result by Morgenstern from 1973, provides an $\Omega(n \log n)$ lower bound for the unnormalized Fourier transform when the constants used in the computation are bounded. The proof uses a potential function related to a determinant. The result does not explain why the normalized Fourier transform (of unit determinant) should be difficult to compute in the same model. Hence, the result is not scale insensitive. More recently, Ailon (2013) showed that if only unitary 2-by-2 gates are used, and additionally no extra memory is allowed, then the normalized Fourier transform requires $\Omega(n\log n)$ steps. This rather limited result is also sensitive to scaling, but highlights the complexity inherent in the Fourier transform arising from introducing entropy, unlike, say, the identity matrix (which is as complex as the Fourier transform using Morgenstern's arguments, under proper scaling). In this work we extend the arguments of Ailon (2013). In the first extension, which is also the main contribution, we provide a lower bound for computing any scaling of the Fourier transform. Our restriction is that, the composition of all gates up to any point must be a well conditioned linear transformation. The lower bound is $\Omega(R^{-1}n\log n)$, where $R$ is the uniform condition number. Second, we assume extra space is allowed, as long as it contains information of bounded norm at the end of the computation. The main technical contribution is an extension of matrix entropy used in Ailon (2013) for unitary matrices to a potential function computable for any matrix, using Shannon entropy on "quasi-probabilities".