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The Forrelation problem is a central problem that demonstrates an exponential separation between quantum and classical capabilities. In this problem, given query access to n-bit Boolean functions f and g, the goal is to estimate the Forrelation function forr(f,g), which measures the correlation between g and the Fourier transform of f. In this work we provide a new linear algebraic perspective on the Forrelation problem, as opposed to prior analytic approaches. We establish a connection between the Forrelation problem and bent Boolean functions and through this connection, analyze an extremal version of the Forrelation problem where the goal is to distinguish between extremal instances of Forrelation, namely (f,g) with forr(f,g) = 1 and forr(f,g) = -1. We show that this problem can be solved with one quantum query and success probability one, yet requires Ω̃(2^{n/4}) classical randomized queries, even for algorithms with a one-third failure probability, highlighting the remarkable power of one exact quantum query. We also study a restricted variant of this problem where the inputs f,g are computable by small classical circuits and show classical hardness under cryptographic assumptions. 

Abstract: The relative-error property testing model was introduced in [Chen et al., 2024] to facilitate the study of property testing for "sparse" Boolean-valued functions, i.e. ones for which only a small fraction of all input assignments satisfy the function. In this framework, the distance from the unknown target function f that is being tested to a function g is defined as Vol(f△g)/Vol(f), where the numerator is the fraction of inputs on which f and g disagree and the denominator is the fraction of inputs that satisfy f. Recent work [Chen et al., 2026] has shown that over the Boolean domain {0,1}ⁿ, any relative-error testing algorithm for the fundamental class of {halfspaces} (i.e. linear threshold functions) must make Ω(log n) oracle calls. In this paper we complement the [Chen et al., 2026] lower bound by showing that halfspaces can be relative-error tested over ℝⁿ under the standard N(0,I_n) Gaussian distribution using a sublinear number of oracle calls - in particular, substantially fewer than would be required for learning. Our results use a wide range of tools including Hermite analysis, Gaussian isoperimetric inequalities, and geometric results on noise sensitivity and surface area. 

In thetrace reconstruction problem, an unknown source stringx∈ {0,1}ⁿis sent through a probabilisticdeletion channelthat independently deletes each bit with probability δ and concatenates the surviving bits, yielding atraceofx. The problem is to reconstructxgiven independent traces. This problem has received much attention in recent years both in the worst-case setting wherexmay be an arbitrary string in {0,1}ⁿ[7,8,10,11,12,23] and in the average-case setting wherexis drawn uniformly at random from {0,1}ⁿ[7,8,12,13,25]. This article studies trace reconstruction in thesmoothed analysissetting, in which a “worst-case” stringx^worstis chosen arbitrarily from {0,1}ⁿ, and then a perturbed versionxofx^worstis formed by independently replacing each coordinate by a uniform random bit with probability σ. The problem is to reconstructxgiven independent traces from it. Our main result is an algorithm that, for any constant perturbation rate 0< σ < 1 and any constant deletion rate 0 < δ < 1, uses poly(n) running time and traces and succeeds with high probability in reconstructing the stringx. This stands in contrast with the worst-case version of the problem, for which\(\text{exp}(\tilde{O}(n^{1/5}))\)is the best known time and sample complexity [8]. Our approach is based on reconstructingxfrom the multiset of its short subwords and is quite different from previous algorithms for either the worst-case or average-case versions of the problem. The heart of our work is a new poly(n)-time procedure for reconstructing the multiset of allO(logn)-length subwords of any source stringx∈ {0,1}ⁿgiven access to traces ofx.