k-Forrelation Optimally Separates Quantum and Classical Query Complexity
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Aaronson and Ambainis (SICOMP `18) showed that any partial function on N bits that can be computed with an advantage δ over a random guess by making q quantum queries, can also be computed classically with an advantage δ/2 by a randomized decision tree making O_q(N^1-1/2qδ^-2) queries. Moreover, they conjectured the k-Forrelation problem – a partial function that can be computed with q = ⌈ k/2 ⌉ quantum queries – to be a suitable candidate for exhibiting such an extremal separation. We prove their conjecture by showing a tight lower bound of Ω(N^1-1/k) for the randomized query complexity of k-Forrelation, where the advantage δ = 2^-O(k). By standard amplification arguments, this gives an explicit partial function that exhibits an O_ϵ(1) vs Ω(N^1-ϵ) separation between bounded-error quantum and randomized query complexities, where ϵ>0 can be made arbitrarily small. Our proof also gives the same bound for the closely related but non-explicit k-Rorrelation function introduced by Tal (FOCS `20). Our techniques rely on classical Gaussian tools, in particular, Gaussian interpolation and Gaussian integration by parts, and in fact, give a more general statement. We show that to prove lower bounds for k-Forrelation against a family of functions, it suffices to bound the ℓ_1-weight of the Fourier coefficients between levels k and (k-1)k. We also prove new interpolation and integration by parts identities that might be of independent interest in the context of rounding high-dimensional Gaussian vectors.
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