Lower Bounds for XOR of Forrelations
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The Forrelation problem, introduced by Aaronson [A10] and Aaronson and Ambainis [AA15], is a well studied problem in the context of separating quantum and classical models. Variants of this problem were used to give exponential separations between quantum and classical query complexity [A10, AA15]; quantum query complexity and bounded-depth circuits [RT19]; and quantum and classical communication complexity [GRT19]. In all these separations, the lower bound for the classical model only holds when the advantage of the protocol (over a random guess) is more than ≈ 1/√(N), that is, the success probability is larger than ≈ 1/2 + 1/√(N). To achieve separations when the classical protocol has smaller advantage, we study in this work the XOR of k independent copies of the Forrelation function (where k≪ N). We prove a very general result that shows that any family of Boolean functions that is closed under restrictions, whose Fourier mass at level 2k is bounded by α^k, cannot compute the XOR of k independent copies of the Forrelation function with advantage better than O(α^k/N^k/2). This is a strengthening of a result of [CHLT19], that gave a similar result for k=1, using the technique of [RT19]. As an application of our result, we give the first example of a partial Boolean function that can be computed by a simultaneous-message quantum protocol of cost (N) (when players share (N) EPR pairs), however, any classical interactive randomized protocol of cost at most õ(N^1/4), has quasipolynomially small advantage over a random guess. We also give the first example of a partial Boolean function that has a quantum query algorithm of cost (N), and such that, any constant-depth circuit of quasipolynomial size has quasipolynomially small advantage over a random guess.
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