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Entanglement Theory Quantum Correlations Quantum Simulation Quantum State Preparation Representation

Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem

arXiv

Authors: Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, Avishay Tal

Year

2020

Paper ID

19685

Status

Preprint

Abstract Read

~2 min

Abstract Words

153

Citations

N/A

Abstract

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f, bullet quad deg(f) = O\(widetilde{deg}(f\)²): The degree of f is at most quadratic in the approximate degree of f. This is optimal as witnessed by the OR function. bullet quad D(f) = O\(Q(f\)⁴): The deterministic query complexity of f is at most quartic in the quantum query complexity of f. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We apply these results to resolve the quantum analogue of the Aanderaa--Karp--Rosenberg conjecture. We show that if f is a nontrivial monotone graph property of an n-vertex graph specified by its adjacency matrix, then Q(f)=Ω(n), which is also optimal. We also show that the approximate degree of any read-once formula on n variables is Θ\(sqrt{n}\).

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This paper contributes to the Quantum Simulation research area in the Quantum Articles archive.
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Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f, bullet quad deg(f) = O(widetildedeg(f)^2): The degree of f is at most quadratic...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2010.12629 [2] arXiv https://arxiv.org/abs/2010.12629 [3] Publisher https://arxiv.org/abs/2010.12629

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