Quantum Search With Generalized Wildcards

TL;DR

The paper introduces a unified framework to analyze quantum query complexity for learning a hidden string under restricted substring queries by relating it to an optimization over odd functions, via the primal negative-weight adversary bound. It shows that the quantum complexity $\mathsf{Q}^{\\mathcal{Q}}(\\oplus)$ equals $\\Theta(\\mathsf{val}_{\\mathcal{Q}})$ and derives tight bounds for natural families of query sets: bounded-size, contiguous blocks, prefixes, and only the full set. These results, together with a Fourier-analytic treatment of XOR-structured adversaries, recover known bounds for wildcard models (reducing $O(\sqrt{n}\log n)$ to $O(\sqrt{n}))$ and establish new upper bounds without SDP duality. The findings have implications for string-recovery problems in biology and illuminate the potential of primal adversary techniques as a practical tool for quantum-query upper bounds.

Abstract

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 $Ω(\sqrt{n})$. 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.

Figures (1)

• Figure 1: A schematic illustration to obtain our framework, which enables us to transform the problem of computing $\mathsf{Q}^{\mathcal{Q}}(\oplus)$, for an arbitrary collection of sets $\mathcal{Q} \subseteq 2^{[n]}$, into the task of analyzing an abstract analytic optimization program $\mathsf{val}_{\mathcal{Q}}$. The solid lines represent exact equalities, whereas the dashed lines represents characterizations up to constants.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Definition 2.2: Positive semi-definiteness
• Theorem 2.3: Rei11
• Lemma 2.5: BC02, also see M18
• Definition 2.7
• Proposition 2.8: o2014analysis
• Corollary 2.9