Quantum Sabotage Complexity
Arjan Cornelissen, Nikhil S. Mande, Subhasree Patro
TL;DR
This work introduces quantum sabotage complexity, the quantum analog of randomized sabotage, by studying sabotage on sabotaged inputs to a Boolean function. It defines four sabotage variants under weak/strong input models and plain vs index outputs, and shows that, in the strong model, the sabotage cost is at most the usual quantum query cost, i.e., $\mathsf{QS}_{\mathsf{str}}(f) = O(\mathsf{Q}(f))$, with stronger bounds for index-output variants. A key general lower bound ties $\mathsf{QS}_{\mathsf{str}}(f)$ to the fractional block sensitivity via $\mathsf{QS}_{\mathsf{str}}(f) = \Omega(\sqrt{\mathsf{fbs}(f)})$, and the Indexing function provides a tight example where $\mathsf{QS}_{\mathsf{str}}^{\mathsf{ind}}(f)$ scales as $\Theta(\mathsf{fbs}(f))$, ruling out a universal $\Theta(\sqrt{\mathsf{fbs}(f)})$ bound. The results connect quantum sabotage costs to standard complexity measures, yielding polynomial relations between $\mathsf{QS}$ and $\mathsf{Q}$ and highlighting the nuanced differences across input/output models. Together, they illuminate how sabotaging information interacts with quantum query strategies and identify sharp separations illustrated by the Indexing function.
Abstract
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 $\mathsf{Q}(f_{\mathsf{sab}})=O(\min\{\mathsf{Q}(f)^{1.5},\sqrt{n}\})$. $\bullet\;\;$ $\mathsf{Q}(f_{\mathsf{sab}}) = Ω(\sqrt{\mathsf{fbs}(f)})$, 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, $\mathsf{Q}(f_{\mathsf{sab}})=Θ(\mathsf{fbs}(f))$, ruling out the possibility that $\mathsf{Q}(f_{\mathsf{sab}})=Θ(\sqrt{\mathsf{fbs}(f)})$ for all $f$.