Approximate degree lower bounds for oracle identification problems

Mark Bun; Nadezhda Voronova

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arXiv · 2303.03921

Approximate degree lower bounds for oracle identification problems

Mark Bun·Nadezhda Voronova

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Abstract

The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically lifts to a quantum communication lower bound for a related function. We introduce a framework for proving approximate degree lower bounds for certain oracle identification problems, where the goal is to recover a hidden binary string $x \in \{0, 1\}^n$ given possibly non-standard oracle access to it. Our lower bounds apply to decision versions of these problems, where the goal is to compute the parity of $x$. We apply our framework to the ordered search and hidden string problems, proving nearly tight approximate degree lower bounds of $\Omega(n/\log^2 n)$ for each. These lower bounds generalize to the weakly unbounded error setting, giving a new quantum query lower bound for the hidden string problem in this regime. Our lower bounds are driven by randomized communication upper bounds for the greater-than and equality functions.

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Mark Bun, Nadezhda Voronova. 2023-03-07. Approximate degree lower bounds for oracle identification problems. https://arxiv.org/abs/2303.03921

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