We study the query complexity of Boolean functions $f: \{0, 1\}^n \rightarrow \{0, 1\}$ in the noisy query model introduced by Feige, Raghavan, Peleg and Upfal [SICOMP 1994]. In this model, an algorithm can adaptively query the bits of an input vector, but each query result is independently flipped with constant probability $p \in (0, 1/2)$; repeated queries are allowed. The noisy query complexity $\mathsf{N}_p(f)$ of a function $f$ is defined as the minimum expected number of queries needed to compute $f(x)$ with error probability at most $1/3$, for the worst case input $x$. We prove a general lower bound on $\mathsf{N}_p(f)$ based on degree statistics of certain subgraphs of the Boolean hypercube. This is the first general lower bound beyond those implied by the simple observation that $\mathsf{N}_p(f)$ is lower bounded by the randomized query complexity. We show that this recovers (up to a constant factor) most previously known lower bounds on the noisy query complexity of Boolean functions, providing a unified framework for understanding these results and simplifying the proofs in several cases. Furthermore, this resolves in the affirmative an open problem of Gu, Li and Xu [COLT 2025] that $\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))$, where $\mathsf{I}(f)$ denotes the total influence of $f$. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.

翻译：我们研究了由Feige、Raghavan、Peleg和Upfal [SICOMP 1994]提出的噪声查询模型下布尔函数 $f: \{0, 1\}^n \rightarrow \{0, 1\}$ 的查询复杂度。在该模型中，算法可以自适应地查询输入向量的比特位，但每次查询结果以恒定概率 $p \in (0, 1/2)$ 独立翻转；允许重复查询。函数 $f$ 的噪声查询复杂度 $\mathsf{N}_p(f)$ 定义为在最坏情况输入 $x$ 下，以不超过 $1/3$ 的错误概率计算 $f(x)$ 所需的最小期望查询次数。我们基于布尔超立方体某些子图的度数统计，证明了 $\mathsf{N}_p(f)$ 的一个通用下界。这是除 $\mathsf{N}_p(f)$ 受随机化查询复杂度下界约束这一简单观察之外的首个通用下界。我们证明该下界（在常数因子内）恢复了此前已知的关于布尔函数噪声查询复杂度的多数下界，为理解这些结果提供了统一框架，并简化了若干情形下的证明。此外，这肯定地解决了Gu、Li和Xu [COLT 2025]的一个开放问题，即 $\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))$，其中 $\mathsf{I}(f)$ 表示 $f$ 的总影响力。我们还应用这一通用下界，为几个新函数获得了噪声查询复杂度的紧界。