Vanishing-Error Approximate Degree and QMA Complexity
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The ϵ-approximate degree of a function f X →{0, 1} is the least degree of a multivariate real polynomial p such that |p(x)-f(x)| ≤ϵ for all x ∈ X. We determine the ϵ-approximate degree of the element distinctness function, the surjectivity function, and the permutation testing problem, showing they are Θ(n^2/3log^1/3(1/ϵ)), Θ̃(n^3/4log^1/4(1/ϵ)), and Θ(n^1/3log^2/3(1/ϵ)), respectively. Previously, these bounds were known only for constant ϵ. 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 Ω(n^1/4). This improves on the previous best lower bound of Ω(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).
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