Given a Boolean function $f:\{0,1\}^n\to\{0,1\}$, the goal in the usual query model is to compute $f$ on an unknown input $x \in \{0,1\}^n$ while minimizing the number of queries to $x$. One can also consider a "distinguishing" problem denoted by $f_{\mathsf{sab}}$: given an input $x \in f^{-1}(0)$ and an input $y \in f^{-1}(1)$, either all differing locations are replaced by a $*$, or all differing locations are replaced by $\dagger$, and an algorithm's goal is to identify which of these is the case while minimizing the number of queries. Ben-David and Kothari [ToC'18] introduced the notion of randomized sabotage complexity of a Boolean function to be the zero-error randomized query complexity of $f_{\mathsf{sab}}$. A natural follow-up question is to understand $\mathsf{Q}(f_{\mathsf{sab}})$, the quantum query complexity of $f_{\mathsf{sab}}$. In this paper, we initiate a systematic study of this. The following are our main results: $\bullet\;\;$ If we have additional query access to $x$ and $y$, then $\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f),\sqrt{n}\})$. $\bullet\;\;$ If an algorithm is also required to output a differing index of a 0-input and a 1-input, then $\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f)^{1.5},\sqrt{n}\})$. $\bullet\;\;$ $\mathsf{Q}(f_{\mathsf{sab}}) = \Omega(\sqrt{\mathsf{fbs}(f)})$, where $\mathsf{fbs}(f)$ denotes the fractional block sensitivity of $f$. By known results, along with the results in the previous bullets, this implies that $\mathsf{Q}(f_{\mathsf{sab}})$ is polynomially related to $\mathsf{Q}(f)$. $\bullet\;\;$ The bound above is easily seen to be tight for standard functions such as And, Or, Majority and Parity. We show that when $f$ is the Indexing function, $\mathsf{Q}(f_{\mathsf{sab}})=\Theta(\mathsf{fbs}(f))$, ruling out the possibility that $\mathsf{Q}(f_{\mathsf{sab}})=\Theta(\sqrt{\mathsf{fbs}(f)})$ for all $f$.

翻译：给定一个布尔函数$f:\{0,1\}^n\to\{0,1\}$，在通常的查询模型中，目标是在最小化对未知输入$x \in \{0,1\}^n$的查询次数的同时计算$f(x)$。我们也可以考虑一个由$f_{\mathsf{sab}}$表示的“区分”问题：给定一个输入$x \in f^{-1}(0)$和一个输入$y \in f^{-1}(1)$，要么将所有不同的位置替换为$*$，要么将所有不同的位置替换为$\dagger$，算法的目标是在最小化查询次数的同时识别出是哪种情况。Ben-David和Kothari [ToC'18]引入了布尔函数的随机破坏复杂性的概念，定义为$f_{\mathsf{sab}}$的零错误随机查询复杂度。一个自然的后续问题是理解$\mathsf{Q}(f_{\mathsf{sab}})$，即$f_{\mathsf{sab}}$的量子查询复杂度。在本文中，我们对此展开了系统性研究。以下是我们主要的研究结果：$\bullet\;\;$ 如果我们对$x$和$y$有额外的查询权限，那么$\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f),\sqrt{n}\})$。$\bullet\;\;$ 如果算法还需要输出一个0输入与1输入之间的不同位置索引，那么$\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f)^{1.5},\sqrt{n}\})$。$\bullet\;\;$ $\mathsf{Q}(f_{\mathsf{sab}}) = \Omega(\sqrt{\mathsf{fbs}(f)})$，其中$\mathsf{fbs}(f)$表示$f$的分数块敏感度。结合已知结果以及前述要点中的结果，这意味着$\mathsf{Q}(f_{\mathsf{sab}})$与$\mathsf{Q}(f)$呈多项式关系。$\bullet\;\;$ 对于诸如And、Or、Majority和Parity等标准函数，上述界限很容易被证明是紧的。我们证明当$f$是索引函数时，$\mathsf{Q}(f_{\mathsf{sab}})=\Theta(\mathsf{fbs}(f))$，这排除了$\mathsf{Q}(f_{\mathsf{sab}})=\Theta(\sqrt{\mathsf{fbs}(f)})$对所有$f$都成立的可能性。