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arXiv:1305.4745 · Chicago Journal of Theoretical Computer Science

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

Nir Ailon

Published May 21, 2013• 1 citation
arXivHTMLPDF

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 $Ω(n \log n)$ 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 $Ω(n\log n)$ 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 $Ω(n\log n)$ 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 $Ω(n\log n)$ conjectured lower bound for computation of Fourier transform is not related to matrix determinant, but rather to a notion of matrix entropy.

DOI
10.48550/arxiv.1305.4745

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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 Ω(nlog⁡n)Ω(n \log n)Ω(nlogn) 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 nn/2n^{n/2}nn/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 Ω(nlog⁡n)Ω(n\log n)Ω(nlogn) steps to compute. In this work we show that in a layered linear circuit model restricted to unitary 2×22\times 22×2 gates, one obtains an Ω(nlog⁡n)Ω(n\log n)Ω(nlogn) 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 Ω(nlog⁡n)Ω(n\log n)Ω(nlogn) conjectured lower bound for computation of Fourier transform is not related to matrix determinant, but rather to a notion of matrix entropy.
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 Ω(nlog⁡n)\Omega(n \log n)Ω(nlogn) 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 Ω(nlog⁡n)\Omega(n\log n)Ω(nlogn) 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 Ω(R−1nlog⁡n)\Omega(R^{-1}n\log n)Ω(R−1nlogn), where RRR 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.
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 Ω(nlog⁡n)Ω(n \log n)Ω(nlogn) 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 Ω(nlog⁡n)Ω(n\log n)Ω(nlogn) 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 Ω(R−1nlog⁡n)Ω(R^{-1}n\log n)Ω(R−1nlogn), where RRR 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".
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.”
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(nlog⁡n)O(n\log n)O(nlogn). From a lower bound perspective, relatively little is known. Ailon shows in 2013 an Ω(nlog⁡n)Ω(n\log n)Ω(nlogn) bound for computing the normalized Fourier Transform assuming only unitary operations on pairs of coordinates is allowed. The goal of this document is to describe a natural open problem that arises from this work, which is related to group theory, and in particular to representation theory.
The complexity of computing the Fourier transform is a longstanding open problem. Very recently, Ailon (2013, 2014, 2015) showed in a collection of papers that, roughly speaking, a speedup of the Fourier transform computation implies numerical ill-condition. The papers also quantify this tradeoff. The main method for proving these results is via a potential function called quasi-entropy, reminiscent of Shannon entropy. The quasi-entropy method opens new doors to understanding the computational complexity of the important Fourier transformation. However, it suffers from various obvious limitations. This paper, motivated by one such limitation, partly overcomes it, while at the same time sheds llight on new interesting, and problems on the intersection of computational complexity and group theory. The paper also explains why this research direction, if fruitful, has a chance of solving much bigger questions about the complexity of the Fourier transform.
We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of Rn\mathbb{R}^nRn with evaluation and membership queries is Ω~(n)\tildeΩ(n)Ω~(n), nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an n×nn \times nn×n real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.