Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

Björklund, Andreas; Kaski, Petteri; Williams, Ryan

doi:10.1007/s00453-018-0513-7

Cited by 13 publications

(26 citation statements)

References 22 publications

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“…Intuitively, the gain in the entire process comes from the fact that in the reduced instances obtained over small finite fields, the evaluation points of interests are quite densely packed together inside a small product set and a standard application of the multidimensional FFT can be used to solve these small field instances quite fast. Another closely related result is a recent work of Björklund, Kaski and Williams [BKW19] who (among other results) give an algorithm for multivariate multipoint evaluation but their time complexity depending polynomially on the field size (and not polynomially on the logarithm of the field size).…”

Section: Algorithms For Multivariate Multipoint Evaluationmentioning

confidence: 90%

“…In a recent work, Björklund, Kaski and Williams [BKW19] also prove new data structures upper bounds for polynomial evaluations for multivariate polynomials over finite fields. These data structures are algebraic and are based on some very neat geometric ideas closely related to the notion of Kakeya sets over finite fields.…”

Section: Data Structures For Polynomial Evaluationmentioning

confidence: 96%

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Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Bhargava¹,

Ghosh²,

Kumar³

et al. 2021

Preprint

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Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are also closely related to fast algorithms for other natural algebraic questions like polynomial factorization and modular composition. And while nearly linear time algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck [BM74], fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans [Uma08] and Kedlaya & Umans [KU11] gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables n is at most d o(1) where the degree of the input polynomial in every variable is less than d. They also stated the question of designing fast algorithms for the large variable case (i.e. n / ∈ d o(1) ) as an open problem.In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field F q of characteristic p which evaluates an n-variate polynomial of degree less than d in each variable on N inputs in time 1) poly(log q, d, p, n) , provided that p is at most d o(1) , and q is at most (exp(exp(exp(• • • (exp(d))))), where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. n / ∈ d o(1) ), this is the first nearly linear time algorithm for this problem over any (large enough) field.Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the following two independently interesting applications.

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Section: Algorithms For Multivariate Multipoint Evaluationmentioning

confidence: 90%

Section: Data Structures For Polynomial Evaluationmentioning

confidence: 96%

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Bhargava¹,

Ghosh²,

Kumar³

et al. 2021

Preprint

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“…Additionally, for both problems, the best known algorithm is exponential and has runtime close to or equaling 2 n . While poly3-NONBALANCED can be solved in poly(n)2 0.9965n time, the best known algorithm for computing the permanent [12]…”

Section: The Permanent and The Problem Per-int-nonzeromentioning

confidence: 99%

“…Unlike poly3-NSETH(a), as far as we are aware there is no known better-than-brute force algorithm ruling out the conjecture for any value b < 1. The algorithm in [12], which is better-than-brute-force by subexponential factors rules out b = 1.…”

Section: Conjecture 2 [Per-int-nseth(b)] Any Nondeterministic Classimentioning

confidence: 99%

How many qubits are needed for quantum computational supremacy?

Dalzell

Harrow

Koh

et al. 2020

Quantum

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Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy (PH) does not collapse, a stronger version of the statement that P≠NP, which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse conjecture. Our first two conjectures poly3-NSETH(a) and per-int-NSETH(b) take specific classical counting problems related to the number of zeros of a degree-3 polynomial in n variables over F2 or the permanent of an n×n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2cn time steps, where c∈{a,b}. A third conjecture poly3-ave-SBSETH(a′) asserts a similar statement about average-case algorithms living in the exponential-time version of the complexity class SBP. We analyze evidence for these conjectures and argue that they are plausible when a=1/2, b=0.999 and a′=1/2.Imposing poly3-NSETH(1/2) and per-int-NSETH(0.999), and assuming that the runtime of a hypothetical quantum circuit simulation algorithm would scale linearly with the number of gates/constraints/optical elements, we conclude that Instantaneous Quantum Polynomial-Time (IQP) circuits with 208 qubits and 500 gates, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and 500 constraints and boson sampling circuits (i.e. linear optical networks) with 98 photons and 500 optical elements are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology. Imposing poly3-ave-SBSETH(1/2), we additionally rule out simulations with constant additive error for IQP and QAOA circuits of the same size. Without the assumption of linearly increasing simulation time, we can make analogous statements for circuits with slightly fewer qubits but requiring 104 to 107 gates.

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“…We would like to thank the anonymous referees for their valuable comments, in particular for pointing to [4] and [1].…”

Section: Acknowledgmentmentioning

confidence: 99%

Conical Kakeya and Nikodym sets in finite fields

Warren

Winterhof

2019

Finite Fields and Their Applications

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A Kakeya set contains a line in each direction. Dvir proved a lower bound on the size of any Kakeya set in a finite field using the polynomial method. We prove analogues of Dvir's result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.)Note that the bound on conical Kakeya sets has been known before, however, without an explicitly given constant which is included in our result and close to being best possible.

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Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants

Cited by 13 publications

References 22 publications

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

How many qubits are needed for quantum computational supremacy?

Conical Kakeya and Nikodym sets in finite fields

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