Given oracle access to a real-valued function on the n-dimensional Boolean cube, how many queries does it take to estimate the squared Euclidean distance to its closest linear function within ε? Our main result is that O(log³(1/ε) ⋅ 1/ε²) queries suffice. Not only is the query complexity independent of n but it is optimal up to the polylogarithmic factor. Our estimator evaluates f on pairs correlated by noise rates chosen to cancel out the low-degree contributions to f while leaving the linear part intact. The query complexity is optimized when the noise rates are multiples of Chebyshev nodes. In contrast, we show that the dependence on n is unavoidable in two closely related settings. For estimation from random samples, Θ(√n/ε + 1/ε²) samples are necessary and sufficient. For agnostically learning a linear approximation with ε mean-square regret under the uniform distribution, Ω(n/√ε) nonadaptively chosen queries are necessary, while O(n/ε) random samples are known to be sufficient (Linial, Mansour, and Nisan). Our upper bounds apply to functions with bounded 4-norm. Our lower bounds apply even to ± 1-valued functions.
Given oracle access to a real-valued function on the n-dimensional Boolean cube, how many queries does it take to estimate the squared Euclidean distance to its closest linear function within ε? Our main result is that O(log³(1/ε) ⋅ 1/ε²) queries suffice. Not only is the query complexity independent of n but it is optimal up to the polylogarithmic factor. Our estimator evaluates f on pairs correlated by noise rates chosen to cancel out the low-degree contributions to f while leaving the linear part intact. The query complexity is optimized when the noise rates are multiples of Chebyshev nodes. In contrast, we show that the dependence on n is unavoidable in two closely related settings. For estimation from random samples, Θ(√n/ε + 1/ε²) samples are necessary and sufficient. For agnostically learning a linear approximation with ε mean-square regret under the uniform distribution, Ω(n/√ε) nonadaptively chosen queries are necessary, while O(n/ε) random samples are known to be sufficient (Linial, Mansour, and Nisan). Our upper bounds apply to functions with bounded 4-norm. Our lower bounds apply even to ± 1-valued functions.
Given a set of n d-dimensional Boolean vectors with the promise that the vectors are chosen uniformly at random with the exception of two vectors that have Pearson-correlation ρ (Hamming distance d · 1-ρ/2), how quickly can one find the two correlated vectors? We present an algorithm which, for any constants ε, ρ >; 0 and d >;>; logn/ρ2, finds the correlated pair with high probability, and runs in time O(n 3ω/4 + ϵ)1.8), where w2-O(ρ)), given by Paturi et al. [15], Locality Sensitive Hashing (LSH) [11] and the Bucketing Codes approach [6]. Applications and extensions of this basic algorithm yield improved algorithms for several other problems: ApproximateClosest Pair: For any sufficiently small constant ϵ >; 0, given n vectors in Rd, our algorithm returns a pair of vectors whose Euclidean distance differs from that of the closest pair by a factor of at most 1+ϵ, and runs in time O(n2-Θ(√ϵ)). The best previous algorithms (including LSH) have runtime O(n2-O(ϵ)). Learning Sparse Parity with Noise: Given samples from an instance of the learning parity with noise problem where each example has length n, the true parity set has size at most kkpoly(1/1-2η)0.8kpoly(1/1-2η). This is the first algorithm with no depenJence on η in the exponent of n, aside from the trivial brute-force algorithm. Learning k-Juntas with Noise: Given uniformly random length n Boolean vectors, together with a label, which is some function of just kkpoly(1/1-2η)0.8kpoly(1/1-2η), 2 which improves on the previous best of >; nk(1-2/2k)poly( 1/1-2η ), from [8]. Learning k-Juntas without Noise:1 Our results for learning sparse parities with noise imply an algorithm for learning juntas without noise with runtime n ω+ϵ/4kpoly(n)0.6kpoly(n), which improves on the runtime n ω+1/ω poly(n) ≈ n0.7kpoly(n) of Mossel n et al. [13].
Let X = (z,, , z,-,) be a sequence of n numbers.For 6 > 0, we say that 5; is an e-approximate median if the number of elements strictly less than zi and the number of elements strictly greater than zi are each less than (1 + 6):.We consider the quantum query complexity of computing an c-approximate median, given the sequence X as an oracle.We prove a lower bound of n(min{t,n}) queries for any quantum algorithm that computes an r-approximate median with any constant probability greater than l/2.We also show how an c-approximate median may be computed with 0( $ log(t) log log( $)) oracle queries, which rep resents an improvement over an earlier algorithm due to Grover [ll, 121.Thus, the lower bound we obtain is essentially optimal.The upper and the lower bound both hold in the comparison tree model as well.Our lower bound result is an application of the polynomial paradigm recently introduced to quantum complexity theory by Be& et ol.[l].The main ingredient in the proof is a polynomial degree lower bound far real multilinear polynomials that "approximate" symmetric partial boolean functions.The degree bound extends a result of Patti[15] and also immediately yields lower bounds for the problems of approximating the kth-smallest element, approximating the mean of a sequence of numbers, and approximately counting the number of ones of a boolean function.All bounds obtained come within a polylogarithmic factor of the optimal (as we show by presenting algorithms where no such optimal or near optimal algorithms were known), thus demonstrating the power of the polynomial method.
We investigate the computational complexity of estimating the operator norm distance T∞(ρ0,ρ1), defined via the operator norm ∥A∥∞=σmax(A), given poly(n)-size state-preparation circuits of n-qubit quantum states ρ0 and ρ1. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states:
1. When one state is pure, we establish an optimal quantum estimator using Θ(1/ε) queries to the state-preparation circuits. Consequently, for constant additive error, say ε=1/5, our estimator runs in poly(n) time. Since the operator norm distance T∞(∣ψ⟩⟨ψ∣,ρ) is exactly half of the trace distance T(∣ψ⟩⟨ψ∣,ρ), our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gily{é}n, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with rank(ρ), which can be exp(n) in general.
2. For general quantum states, we also provide a quantum estimator using O(1/ε3/2) queries to the state-preparation circuits, which shows that the corresponding promise problem is BQP-complete and improves the QMA upper bound sketched by Liu and Wang (ESA 2025). Together with an Ω(1/ε) quantum query complexity lower bound, this leaves only square-root room for improvement.
The key intuition behind our estimators is that, when one state is pure, the pure state ∣ψ⟩ has overlap at least 1/2 with the top unit eigenvector of ∣ψ⟩⟨ψ∣−ρ, reflecting a structural feature specific to the operator norm distance.
The Total Influence ( 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, 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 ± ε ) by performing O (√ n I [ f ] poly(1/ ε )) queries. We also prove a lower bound of Ω (√ n log n · I [ f ]) on the query complexity of any constant factor approximation algorithm for this problem (which holds for I [ f ]= Ω (1)), 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 ( n I [ f ]), which matches the complexity of a simple sampling algorithm.
In this work, we develop new insights into the fundamental problem of convexity testing of real-valued functions over the domain [n]. Specifically, we present a nonadaptive algorithm that, given inputs ε ∊ (0,1), s ∊ ℕ, and oracle access to a function, ε-tests convexity in O(log(s)/ε), where s is an upper bound on the number of distinct discrete derivatives of the function. We also show that this bound is tight. Since s ≤ n, our query complexity bound is at least as good as that of the optimal convexity tester (Ben Eliezer; ITCS 2019) with complexity ; our bound is strictly better when s = o(n). The main contribution of our work is to appropriately parameterize the complexity of convexity testing to circumvent the worst-case lower bound (Belovs et al.; SODA 2020) of expressed in terms of the input size and obtain a more efficient algorithm.
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.