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TR20-127 | 21st August 2020 21:46

#### $k$-Forrelation Optimally Separates Quantum and Classical Query Complexity

TR20-127
Authors: Nikhil Bansal, Makrand Sinha
Publication: 21st August 2020 22:02
Aaronson and Ambainis (SICOMP '18) showed that any partial function on $N$ bits that can be computed with an advantage $\delta$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $\delta/2$ by a randomized decision tree making ${O}_q(N^{1-\frac{1}{2q}}\delta^{-2})$ queries. Moreover, they conjectured the $k$-Forrelation problem --- a partial function that can be computed with $q = \lceil k/2 \rceil$ quantum queries --- to be a suitable candidate for exhibiting such an extremal separation.
We prove their conjecture by showing a tight lower bound of $\widetilde{\Omega}_k(N^{1-1/k})$ for the randomized query complexity of $k$-Forrelation, where the advantage $\delta = 1/\mathrm{polylog}^k(N)$ and $\widetilde{\Omega}_k$ hides $\mathrm{polylog}^k(N)$ factors. Our proof relies on classical Gaussian tools, in particular, Gaussian interpolation and Gaussian integration by parts, and in fact, shows a more general statement, that to prove lower bounds for $k$-Forrelation against a family of functions, it suffices to bound the $\ell_1$-weight of the Fourier coefficients at levels $k, 2k, 3k, \ldots, (k-1)k$ for functions in the family.