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• W2973719433 abstract "The $epsilon$-approximate degree of a function $fcolon X to {0, 1}$ is the least degree of a multivariate real polynomial $p$ such that $|p(x)-f(x)| leq epsilon$ for all $x in X$. We determine the $epsilon$-approximate degree of the element distinctness function, the surjectivity function, and the permutation testing problem, showing they are $Theta(n^{2/3} log^{1/3}(1/epsilon))$, $tildeTheta(n^{3/4} log^{1/4}(1/epsilon))$, and $Theta(n^{1/3} log^{2/3}(1/epsilon))$, respectively. Previously, these bounds were known only for constant $epsilon.$ We also derive a connection between vanishing-error approximate degree and quantum Merlin--Arthur (QMA) query complexity. We use this connection to show that the QMA complexity of permutation testing is $Omega(n^{1/4})$. This improves on the previous best lower bound of $Omega(n^{1/6})$ due to Aaronson (Quantum Information & Computation, 2012), and comes somewhat close to matching a known upper bound of $O(n^{1/3})$." @default.
• W2973719433 created "2019-09-26" @default.
• W2973719433 creator A5079755716 @default.
• W2973719433 creator A5080601320 @default.
• W2973719433 date "2019-09-16" @default.
• W2973719433 modified "2023-09-27" @default.
• W2973719433 title "Vanishing-Error Approximate Degree and QMA Complexity" @default.
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