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arXiv:2502.04632

Tight Bounds for Noisy Computation of High-Influence Functions, Connectivity, and Threshold

Yuzhou Gu, Xin Jun Li, Yinzhan Xu

Published Feb 7, 2025• 0 citations
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Abstract

In the noisy query model, the (binary) return value of every query (possibly repeated) is independently flipped with some fixed probability $p \in (0, 1/2)$. In this paper, we obtain tight bounds on the noisy query complexity of several fundamental problems. Our first contribution is to show that any Boolean function with total influence $Ω(n)$ has noisy query complexity $Θ(n\log n)$. Previous works often focus on specific problems, and it is of great interest to have a characterization of noisy query complexity for general functions. Our result is the first noisy query complexity lower bound of this generality, beyond what was known for random Boolean functions [Reischuk and Schmeltz, FOCS 1991]. Our second contribution is to prove that Graph Connectivity has noisy query complexity $Θ(n^2 \log n)$. In this problem, the goal is to determine whether an undirected graph is connected using noisy edge queries. While the upper bound can be achieved by a simple algorithm, no non-trivial lower bounds were known prior to this work. Last but not least, we determine the exact number of noisy queries (up to lower order terms) needed to solve the $k$-Threshold problem and the Counting problem. The $k$-Threshold problem asks to decide whether there are at least $k$ ones among $n$ bits, given noisy query access to the bits. We prove that $(1\pm o(1)) \frac{n\log (\min\{k,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}$ queries are both sufficient and necessary to achieve error probability $δ= o(1)$. Previously, such a result was only known when $\min\{k,n-k+1\}=o(n)$ [Wang, Ghaddar, Zhu and Wang, arXiv 2024]. We also show a similar $(1\pm o(1)) \frac{n\log (\min\{k+1,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}$ bound for the Counting problem, where one needs to count the number of ones among $n$ bits given noisy query access and $k$ denotes the answer.

DOI
10.48550/arxiv.2502.04632

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The {\em Total Influence} ({\em Average Sensitivity) of a discrete function is one of its fundamental measures. We study the problem of approximating the total influence of a monotone Boolean function \ifnum\plusminus=1 $f: \{\pm1\}^n \longrightarrow \{\pm1\}$, \else $f: \bitset^n \to \bitset$, \fi which we denote by $I[f]$. We present a randomized algorithm that approximates the influence of such functions to within a multiplicative factor of $(1\pm \eps)$ by performing $O(\frac{\sqrt{n}\log n}{I[f]} \poly(1/\eps)) $ queries. % \mnote{D: say something about technique?} We also prove a lower bound of % $Ω(\frac{\sqrt{n/\log n}}{I[f]})$ $Ω(\frac{\sqrt{n}}{\log n \cdot I[f]})$ on the query complexity of any constant-factor approximation algorithm for this problem (which holds for $I[f] = Ω(1)$), % and $I[f] = O(\sqrt{n}/\log n)$), hence showing that our algorithm is almost optimal in terms of its dependence on $n$. For general functions we give a lower bound of $Ω(\frac{n}{I[f]})$, which matches the complexity of a simple sampling algorithm.

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In the noisy query model, the (binary) return value of every query (possibly repeated) is independently flipped with some fixed probability p∈(0,1/2)p \in (0, 1/2)p∈(0,1/2). In this paper, we obtain tight bounds on the noisy query complexity of several fundamental problems. Our first contribution is to show that any Boolean function with total influence Ω(n)Ω(n)Ω(n) has noisy query complexity Θ(nlog⁡n)Θ(n\log n)Θ(nlogn). Previous works often focus on specific problems, and it is of great interest to have a characterization of noisy query complexity for general functions. Our result is the first noisy query complexity lower bound of this generality, beyond what was known for random Boolean functions [Reischuk and Schmeltz, FOCS 1991]. Our second contribution is to prove that Graph Connectivity has noisy query complexity Θ(n2log⁡n)Θ(n^2 \log n)Θ(n2logn). In this problem, the goal is to determine whether an undirected graph is connected using noisy edge queries. While the upper bound can be achieved by a simple algorithm, no non-trivial lower bounds were known prior to this work. Last but not least, we determine the exact number of noisy queries (up to lower order terms) needed to solve the kkk-Threshold problem and the Counting problem. The kkk-Threshold problem asks to decide whether there are at least kkk ones among nnn bits, given noisy query access to the bits. We prove that (1±o(1))nlog⁡(min⁡{k,n−k+1}/δ)(1−2p)log⁡1−pp(1\pm o(1)) \frac{n\log (\min\{k,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}(1±o(1))(1−2p)logp1−p​nlog(min{k,n−k+1}/δ)​ queries are both sufficient and necessary to achieve error probability δ=o(1)δ= o(1)δ=o(1). Previously, such a result was only known when min⁡{k,n−k+1}=o(n)\min\{k,n-k+1\}=o(n)min{k,n−k+1}=o(n) [Wang, Ghaddar, Zhu and Wang, arXiv 2024]. We also show a similar (1±o(1))nlog⁡(min⁡{k+1,n−k+1}/δ)(1−2p)log⁡1−pp(1\pm o(1)) \frac{n\log (\min\{k+1,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}(1±o(1))(1−2p)logp1−p​nlog(min{k+1,n−k+1}/δ)​ bound for the Counting problem, where one needs to count the number of ones among nnn bits given noisy query access and kkk denotes the answer.
We study the query complexity of Boolean functions f:{0,1}n→{0,1}f: \{0, 1\}^n \rightarrow \{0, 1\}f:{0,1}n→{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∈(0,1/2)p \in (0, 1/2)p∈(0,1/2); repeated queries are allowed. The noisy query complexity Np(f)\mathsf{N}_p(f)Np​(f) of a function fff is defined as the minimum expected number of queries needed to compute f(x)f(x)f(x) with error probability at most 1/31/31/3, for the worst case input xxx. We prove a general lower bound on Np(f)\mathsf{N}_p(f)Np​(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 Np(f)\mathsf{N}_p(f)Np​(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 Np(f)=Ω(I(f)log⁡I(f))\mathsf{N}_p(f) = Ω(\mathsf{I}(f) \log \mathsf{I}(f))Np​(f)=Ω(I(f)logI(f)), where I(f)\mathsf{I}(f)I(f) denotes the total influence of fff. We also apply our general lower bound to obtain tight bounds on the noisy query complexity for several new functions.
Let THk\mathsf{TH}_kTHk​ denote the kkk-out-of-nnn threshold function: given nnn input Boolean variables, the output is 111 if and only if at least kkk of the inputs are 111. We consider the problem of computing the THk\mathsf{TH}_kTHk​ function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability p∈(0,1/2)p \in (0,1/2)p∈(0,1/2). As our main result, we show that it is sufficient to use (1+o(1))nlog⁡mδDKL(p∥1−p)(1+o(1)) \frac{n\log \frac{m}δ}{D_{\mathsf{KL}}(p \| 1-p)}(1+o(1))DKL​(p∥1−p)nlogδm​​ queries in expectation to compute the THk\mathsf{TH}_kTHk​ function with a vanishing error probability δ=o(1)δ= o(1)δ=o(1), where m≜min⁡{k,n−k+1}m\triangleq \min\{k,n-k+1\}m≜min{k,n−k+1} and DKL(p∥1−p)D_{\mathsf{KL}}(p \| 1-p)DKL​(p∥1−p) denotes the Kullback-Leibler divergence between Bern(p)\mathsf{Bern}(p)Bern(p) and Bern(1−p)\mathsf{Bern}(1-p)Bern(1−p) distributions. Conversely, we show that any algorithm achieving an error probability of δ=o(1)δ= o(1)δ=o(1) necessitates at least (1−o(1))(n−m)log⁡mδDKL(p∥1−p)(1-o(1))\frac{(n-m)\log\frac{m}δ}{D_{\mathsf{KL}}(p \| 1-p)}(1−o(1))DKL​(p∥1−p)(n−m)logδm​​ queries in expectation. The upper and lower bounds are tight when m=o(n)m=o(n)m=o(n), and are within a multiplicative factor of nn−m\frac{n}{n-m}n−mn​ when m=Θ(n)m=Θ(n)m=Θ(n). In particular, when k=n/2k=n/2k=n/2, the THk\mathsf{TH}_kTHk​ function corresponds to the MAJORITY\mathsf{MAJORITY}MAJORITY function, in which case the upper and lower bounds are tight up to a multiplicative factor of two. Compared to previous work, our result tightens the dependence on ppp in both the upper and lower bounds.
Noisy Computing of the OR\mathsf{OR}OR and MAX\mathsf{MAX}MAX Functions
We consider the problem of computing a function of nnn variables using noisy queries, where each query is incorrect with some fixed and known probability p∈(0,1/2)p \in (0,1/2)p∈(0,1/2). Specifically, we consider the computation of the OR\mathsf{OR}OR function of nnn bits (where queries correspond to noisy readings of the bits) and the MAX\mathsf{MAX}MAX function of nnn real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of (1±o(1))nlog⁡1δDKL(p∥1−p) (1 \pm o(1)) \frac{n\log \frac{1}δ}{D_{\mathsf{KL}}(p \| 1-p)} (1±o(1))DKL​(p∥1−p)nlogδ1​​ is both sufficient and necessary to compute both functions with a vanishing error probability δ=o(1)δ= o(1)δ=o(1), where DKL(p∥1−p)D_{\mathsf{KL}}(p \| 1-p)DKL​(p∥1−p) denotes the Kullback-Leibler divergence between Bern(p)\mathsf{Bern}(p)Bern(p) and Bern(1−p)\mathsf{Bern}(1-p)Bern(1−p) distributions. Compared to previous work, our results tighten the dependence on ppp in both the upper and lower bounds for the two functions.
Subgraph detection asks whether an nnn-vertex graph, accessed through queries to its adjacency matrix, contains a copy of a fixed graph HHH. We prove the first unconditional superlinear lower bounds on the bounded-error quantum query complexity of this problem, answering a longstanding open question. A copy of HHH is a certificate of constant size, so the adversary method with nonnegative weights cannot prove superlinear lower bounds. For every fixed r≥4r\ge 4r≥4, detecting the clique KrK_rKr​ requires nλr−o(1)n^{λ_r-o(1)}nλr​−o(1) queries, where λ4=19/18λ_4=19/18λ4​=19/18, the exponents λrλ_rλr​ increase strictly with rrr, and λr≥2−42/r+O(1/r)λ_r\ge 2-4\sqrt{2/r}+O(1/r)λr​≥2−42/r​+O(1/r). More generally, we prove superlinear lower bounds for detecting every fixed connected graph HHH with chromatic number c≥4c\ge 4c≥4. These bounds approach quadratic as ccc grows: for sufficiently large ccc, detection requires n2−O(log⁡log⁡c/c)−o(1)n^{2-O(\sqrt{\log\log c/c})-o(1)}n2−O(loglogc/c​)−o(1) queries. Chromatic number alone does not characterize the quantum query complexity of subgraph detection: we show that detecting the complete bipartite graph Kr,rK_{r,r}Kr,r​ requires nβr−o(1)n^{β_r-o(1)}nβr​−o(1) queries, where β10=181/180β_{10}=181/180β10​=181/180 and βr≥2−O(1/r)β_r\ge 2-O(1/\sqrt{r})βr​≥2−O(1/r​). Our main technical result is a lower bound for finding an all-ones certificate from a known family when the input bits are sampled independently. Its proof combines Zhandry's compressed oracle [CRYPTO 2019] with conditioning on a randomly planted certificate, adapting an argument of Belovs [FOCS 2026]. Our hard instances are built from graphs containing many copies of the desired subgraph with limited overlap. For cliques, we use a construction of Gowers and Janzer [CPC 2021]; for complete bipartite graphs, we use a random construction.
The edge connectivity of a simple graph is the least number of edges whose removal disconnects the graph. We study the quantum query complexity of computing the edge connectivity of a simple graph with nnn vertices and mmm edges in both the adjacency matrix and adjacency array models. For classical randomized algorithms, Ω(n2)\Omega(n^2)Ω(n2) and Ω(m)\Omega(m)Ω(m) lower bounds are known for these models, respectively, showing that the trivial algorithm is optimal in terms of query complexity for both cases. In contrast, we show upper bounds of O~(n3/2)\tilde O(n^{3/2})O~(n3/2) and O~(mn)\tilde O(\sqrt{mn})O~(mn​) on the quantum query complexity of computing edge connectivity in the adjacency matrix and adjacency array models, respectively. Our upper bound is tight up to logarithmic factors for the adjacency matrix model, while for the adjacency array model the best quantum query lower bound we know is Ω(n)\Omega(n)Ω(n). Our upper bounds follow by combining a lemma on the structure of near-minimum cuts due to Rubinstein, Schramm and Weinberg (ITCS 2018) with a quantum sparsification algorithm by Apers and de Wolf (FOCS 2020). For a weighted graph, instead of edge connectivity the relevant notion is the weight of a minimum cut, i.e. the minimum total weight of a set of edges whose removal disconnects the graph. For weighted graphs we show that in the worst case Ω(n2)\Omega(n^2)Ω(n2) queries can be needed to compute the weight of a minimum cut by a quantum algorithm in both the adjacency matrix and adjacency array models, thus no quantum speedup is possible in this case.
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