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Classical Lower Bounds from Quantum Upper Bounds

“…This lower bound is also tight up to log factors due to a matching upper bound of Sherstov [She13c]. This resolves the symmetric-composition question for approximate degree.…”

“…These lower bounds were proved using a linear programming formulation of approximate degree, and exploited certain properties of the dual polynomial for the OR function. In contrast to these approaches using dual polynomials, our OR-composition result for approximate degree uses completely different techniques and is more general: This lower bound is tight due to a matching upper bound of Sherstov [She13c]. This resolves the OR-composition question for approximate degree.…”

“…We would now like to compose this polynomial with the assumed polynomials that allow us to compute the OR of a subset of the functions f . However, since the polynomials we wish to compose are approximating polynomials, they do not straightforwardly compose as expected, and to make this work, we use Sherstov's robust polynomial construction [She13c]. Finally, by composing these polynomials of degree T and degree O( √ n), we get a polynomial of degree O(T √ n) for computing the parity of all the functions f , i.e., we have shown that deg(XOR n •f ) = O(T √ n).…”

“…While this is the key conceptual step needed for the generalization, working with partial functions presents several technical challenges. One of the main challenges corresponds to the robustness of polynomials, for which we used Sherstov's robust polynomial construction [She13c] previously.…”

Near-optimal lower bounds on the threshold degree and sign-rank of AC 0

“…To illustrate the issue, consider arbitrary functions f : {−1, +1} n1 → {−1, +1} and g : {−1, +1} n2 → {−1, +1} with 1/3-approximate degrees n α1 1 and n α2 2 , respectively, for some 0 < α 1 < 1 and 0 < α 2 < 1. It is well-known [48] that the composed function f • g on n 1 n 2 variables has 1/3approximate degree O(n α1 1 n α2 2 ) = O(n 1 n 2 ) max{α1,α2} . This means that relative to the new number of variables, the block-composed function f • g is asymptotically no harder to approximate to bounded error than the constituent functions f and g. In particular, one cannot use block-composition to transform functions on n bits with 1/3-approximate degree at most n α into functions on N n bits with 1/3-approximate degree ω(N α ).…”