
Quantum Implications of Huang's Sensitivity Theorem
Based on the recent breakthrough of Huang (2019), we show that for any t...
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On the Probabilistic Degree of an nvariate Boolean Function
Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}^n→{...
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Algorithmic Polynomials
The approximate degree of a Boolean function f(x_1,x_2,...,x_n) is the m...
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Quantum LogApproximateRank Conjecture is also False
In a recent breakthrough result, Chattopadhyay, Mande and Sherif [ECCC T...
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A Quantum Query Complexity Trichotomy for Regular Languages
We present a trichotomy theorem for the quantum query complexity of regu...
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Symmetries, graph properties, and quantum speedups
Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symm...
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Quantum algorithms and approximating polynomials for composed functions with shared inputs
We give new quantum algorithms for evaluating composed functions whose i...
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Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem
Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f, ∙ deg(f) = O(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. ∙ 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 nvertex graph specified by its adjacency matrix, then Q(f)=Ω(n), which is also optimal. We also show that the approximate degree of any readonce formula on n variables is Θ(√(n)).
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