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.
We study the query complexity of Boolean functions 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); repeated queries are allowed. The noisy query complexity Np(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 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) 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)logI(f)), where 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.
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). 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(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(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 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±o(1))(1−2p)logp1−pnlog(min{k,n−k+1}/δ) 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±o(1))(1−2p)logp1−pnlog(min{k+1,n−k+1}/δ) 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.
Given a Boolean function f:{0,1}n→{0,1}, the goal in the usual query model is to compute f on an unknown input x∈{0,1}n while minimizing the number of queries to x. One can also consider a "distinguishing" problem denoted by fsab: given an input x∈f−1(0) and an input y∈f−1(1), either all differing locations are replaced by a ∗, or all differing locations are replaced by †, 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 fsab. A natural follow-up question is to understand Q(fsab), the quantum query complexity of fsab. In this paper, we initiate a systematic study of this. The following are our main results: ∙ If we have additional query access to x and y, then Q(fsab)=O(min{Q(f),n}). ∙ If an algorithm is also required to output a differing index of a 0-input and a 1-input, then Q(fsab)=O(min{Q(f)1.5,n}). ∙Q(fsab)=Ω(fbs(f)), where fbs(f) denotes the fractional block sensitivity of f. By known results, along with the results in the previous bullets, this implies that Q(fsab) is polynomially related to Q(f). ∙ 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, Q(fsab)=Θ(fbs(f)), ruling out the possibility that Q(fsab)=Θ(fbs(f)) for all f.
We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in the quantum case for any eps>0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.
Let THk denote the k-out-of-n threshold function: given n input Boolean variables, the output is 1 if and only if at least k of the inputs are 1. We consider the problem of computing the THk function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability p∈(0,1/2). As our main result, we show that it is sufficient to use (1+o(1))DKL(p∥1−p)nlogδm queries in expectation to compute the THk function with a vanishing error probability δ=o(1), where m≜min{k,n−k+1} and DKL(p∥1−p) denotes the Kullback-Leibler divergence between Bern(p) and Bern(1−p) distributions. Conversely, we show that any algorithm achieving an error probability of δ=o(1) necessitates at least (1−o(1))DKL(p∥1−p)(n−m)logδm queries in expectation. The upper and lower bounds are tight when m=o(n), and are within a multiplicative factor of n−mn when m=Θ(n). In particular, when k=n/2, the THk function corresponds to the 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 p in both the upper and lower bounds.
The noise sensitivity of a Boolean function f: {0,1}^n - > {0,1} is one of its fundamental properties. For noise parameter delta, the noise sensitivity is denoted as NS_{delta}[f]. This quantity is defined as follows: First, pick x = (x_1,...,x_n) uniformly at random from {0,1}^n, then pick z by flipping each x_i independently with probability delta. NS_{delta}[f] is defined to equal Pr [f(x) != f(z)]. Much of the existing literature on noise sensitivity explores the following two directions: (1) Showing that functions with low noise-sensitivity are structured in certain ways. (2) Mathematically showing that certain classes of functions have low noise sensitivity. Combined, these two research directions show that certain classes of functions have low noise sensitivity and therefore have useful structure. The fundamental importance of noise sensitivity, together with this wealth of structural results, motivates the algorithmic question of approximating NS_{delta}[f] given an oracle access to the function f. We show that the standard sampling approach is essentially optimal for general Boolean functions. Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply transformed into a monotone function (for example the class of unate functions consists of all the functions that can be made monotone by reorienting some of their coordinates [O'Donnell, 2014]). Specifically, we study the algorithmic problem of approximating NS_{delta}[f] for monotone f, given the promise that NS_{delta}[f] >= 1/n^{C} for constant C, and for delta in the range 1/n <= delta <= 1/2. For such f and delta, we give a randomized algorithm performing O((min(1,sqrt{n} delta log^{1.5} n))/(NS_{delta}[f]) poly (1/epsilon)) queries and approximating NS_{delta}[f] to within a multiplicative factor of (1 +/- epsilon). Given the same constraints on f and delta, we also prove a lower bound of Omega((min(1,sqrt{n} delta))/(NS_{delta}[f] * n^{xi})) on the query complexity of any algorithm that approximates NS_{delta}[f] to within any constant factor, where xi can be any positive constant. Thus, our algorithm’s query complexity is close to optimal in terms of its dependence on n. We introduce a novel descending-ascending view of noise sensitivity, and use it as a central tool for the analysis of our algorithm. To prove lower bounds on query complexity, we develop a technique that reduces computational questions about query complexity to combinatorial questions about the existence of "thin" functions with certain properties. The existence of such "thin" functions is proved using the probabilistic method. These techniques also yield new lower bounds on the query complexity of approximating other fundamental properties of Boolean functions: the total influence and the bias.
One of the main reasons for query model's prominence in quantum complexity is the presence of concrete lower bounding techniques: polynomial method and adversary method. There have been considerable efforts to not just give lower bounds using these methods but even to compare and relate them. We explore the value of these bounds on quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions. We show (using the recently introduced spectral sensitivity) that both these bounds (positive adversary and approximate degree) give the same value for every total symmetric Boolean function. We also look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. In addition, we study how large certificate complexity and block sensitivity can be as compared to sensitivity (even up to constant factors) for symmetric functions. We show tight separations, i.e., give upper bound on possible separations and construct functions achieving the same.