Quantum Search With Generalized Wildcards

In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string $x \in \{-1,1\}^n$. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost $O(\sqrt{n} \log n)$ and a near-matching lower bound of $\Omega(\sqrt{n})$. Belovs [Comput.~Comp.'15] subsequently showed a tight $O(\sqrt{n})$ upper bound. We consider a natural generalization of this problem, parametrized by a subset $\cal{Q} \subseteq 2^{[n]}$, where an algorithm may test whether $x_S = b$ for an arbitrary $S \in \cal{Q}$ and $b \in \{-1,1\}^S$ of its choice, at unit cost. We show near-tight bounds when $\cal{Q}$ is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set. All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning $x$ is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions $f : \{-1,1\}^n \to \mathbb{R}$ the ratio of the maximum value of $f$ to the maximum (over $T \in \cal{Q}$) standard deviation of $f$ on a subcube whose free variables are exactly $T$.' To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.