An Omega((n log n)/R) Lower Bound for Fourier Transform Computation in the R-Well Conditioned Model

Obtaining a nontrivial (superlinear) lower bound for computation of the Fourier transform in the linear circuit model has been a long-standing open problem for more than 40 years. An early result by Morgenstern from 1973, provides an Ω( 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. That result does not explain why the normalized Fourier transform (of unit determinant) should be difficult to compute in the same model. Hence, it 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 Ω( 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). This work improves on Ailon [2013] in two ways: First, we eliminate the scaling restriction and provide a lower bound for computing any scaling of the Fourier transform. Second, we allow the computational model to use extra memory. Our restriction is that the composition of all gates up to any point must be a well- conditioned linear transformation. The lower bound is Ω( R −1 n log n ), where R is the uniform condition number. Well-conditioned is a natural requirement for algorithms accurately computing linear transformations on machine architectures of bounded word size. Hence, this result can be seen as a tradeoff between speed and accuracy. The main technical contribution is an extension of matrix entropy used in Ailon [2013] for unitary matrices to a potential function computable for any invertible matrix, using “quasi-entropy” of “quasi-probabilities.”