Higher-order Fourier Analysis and Applications

Hatami, Hamed; Hatami, Pooya; Lovett, Shachar

doi:10.1561/9781680835939

Cited by 7 publications

(15 citation statements)

References 86 publications

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“…These are the so-called "degreestructural properties". A simple extension of their result [24,Theorem 16.3] implies that a larger class of "homogeneous degree-structural properties" are linear-invariant, linear subspace-hereditary (but not affine-invariant and not affine-subspace-hereditary), and locally characterized, and thus these properties are testable by our main theorem. As an example, one can test whether a function F n p → F p can be written as A 2 + B 2 where both A and B are homogeneous polynomials of some given degree d.…”

Section: Introductionmentioning

confidence: 75%

“…Note that our proof is fairly "hands-off" in the sense that the reader does not need to delve deeply into the details of non-classical polynomials to understand our proof as long as they are willing to take Theorem 3.10 and Theorem 4.1 on faith. We include the necessary definitions to state these result here (see the book [24] for a more thorough presentation). Definition 3.1.…”

Section: Preliminaries On Higher Order Fourier Analysismentioning

confidence: 99%

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Testing linear-invariant properties

Tidor,

Zhao

2019

Preprint

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We prove that all linear-invariant, linear-subspace hereditary, locally characterized properties are proximity-oblivious testable (with one-sided error and constant query-complexity). In other words, we show that we can distinguish functions F n p → [R] satisfying a given property from those that are ǫ-far from satisfying the property as long as the property is definable by restrictions to bounded dimension subspaces.This result can be equivalently stated as an induced arithmetic removal lemma: given a set of colored patternsfor every ǫ > 0 there exists δ > 0 such that if a function f : F n p → [R] has density at most δ of each of the prescribed patterns, then f can be made free of these patterns by recoloring at most ǫp n points.The proof of this result uses two main techniques. The first builds upon a long line of work which applies regularity methods, including results from higher order Fourier analysis, to prove results on property testing and removal lemmas. The second, the main innovation of this work, is a novel recoloring technique that allows us to handle an important obstacle encountered by previous works in the arithmetic setting. Roughly speaking, this obstacle is the inability of regularity methods to regularize functions in a neighborhood of the origin.

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Section: Introductionmentioning

confidence: 75%

Section: Preliminaries On Higher Order Fourier Analysismentioning

confidence: 99%

Testing linear-invariant properties

Tidor,

Zhao

2019

Preprint

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“…However, f U k is increasing in k (see eg. Claim 6.2.2 in [6]), therefore f U d ≥ |bias χ (P)| ≥ ǫ. The result is now immediate from Theorem 1.5.…”

Section: Bias and Rank Of Polynomialsmentioning

confidence: 93%

Polynomial bound for the partition rank vs the analytic rank of tensors

Janzer

2019

Preprint

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A tensor defined over a finite field F has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order d tensor has partition rank 1 if it can be written as a product of two tensors of order less than d, and it has partition rank at most k if it can be written as a sum of k tensors of partition rank 1. In this paper, we prove that if the analytic rank of an order d tensor is at most r, then its partition rank is at most f (r, d, |F|), where, for fixed d and F, f is a polynomial in r. This is an improvement of a recent result of the author, where he obtained a tower-type bound. Prior to our work, the best known bound was an Ackermann-type function in r and d, though it did not depend on F. It follows from our results that a biased polynomial has low rank; there too we obtain a polynomial dependence improving the previously known Ackermann-type bound.A similar polynomial bound for the partition rank was obtained independently and simultaneously by Milićević.

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“…In this section we introduce a notion of rank for nonclassical polynomials. The main reference is again [HHL19]. Definition 3.9.…”

Section: Rank Of Nonclassical Polynomialsmentioning

confidence: 99%

“…This current work is yet another example. Higher-order Fourier analysis has already found many applications in classical theoretical computer science, such as in property testing, coding theory and complexity theory [HHL19].…”

Section: Introductionmentioning

confidence: 99%

Stabilizer rank and higher-order Fourier analysis

Labib

2021

Preprint

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We establish a link between stabilizer states, stabilizer rank and higher-order Fourier analysisa still-developing area of mathematics that grew out of Gowers's celebrated Fourier-analytic proof of Szemerédi's theorem [Gow98]. We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of F n p where p is the dimension of the qudit) which are fundamental objects in higher-order Fourier analysis. This allows us to import tools from this theory to analyze the stabilizer rank of quantum states. Quite recently, in [PSV21] it was shown that the n-qubit magic state has stabilizer rank Ω(n). Here we show that the qudit analogue of the n-qubit magic state has stabilizer rank Ω(n), generalizing their result to qudits of any prime dimension. Our proof techniques use explicitly tools from higher-order Fourier analysis. We believe this example motivates the further exploration of applications of higher-order Fourier analysis in quantum information theory.

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Higher-order Fourier Analysis and Applications

Cited by 7 publications

References 86 publications

Testing linear-invariant properties

Testing linear-invariant properties

Polynomial bound for the partition rank vs the analytic rank of tensors

Stabilizer rank and higher-order Fourier analysis

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