We study the number of uniform quantum examples needed to exactly learn a Boolean function with at most k nonzero Fourier coefficients. We give a learner with near-optimal O(k) sample complexity independent of the input length n. The learner uses only separate measurements of individual examples. This improves the previous O(k3/2log2k) upper bound (Arunachalam, Chakraborty, Lee, Paraashar, and de Wolf, 2021). The main ingredient is a candidate set formed by symmetric differences of subsets from independent groups of Fourier samples. A multilinear majority polynomial combines unbiased random approximations into an accurate witness whose Fourier support lies in this set. Granularity then forces the set to contain the entire Fourier support, including subsets never directly sampled. For comparison, we derive the tight classical sample complexity Θ(nk) by combining known sparse-recovery and lower-bound results.
We prove a new lower bound on the parity decision tree complexity D⊕(f) of a Boolean function f. Namely, granularity of the Boolean function f is the smallest k such that all Fourier coefficients of f are integer multiples of 1/2k. We show that D⊕(f)≥k+1. This lower bound is an improvement of lower bounds through the sparsity of f and through the degree of f over F2. Using our lower bound we determine the exact parity decision tree complexity of several important Boolean functions including majority and recursive majority. For majority the complexity is n−B(n)+1, where B(n) is the number of ones in the binary representation of n. For recursive majority the complexity is 2n+1. Finally, we provide an example of a function for which our lower bound is not tight. Our results imply new lower bound of n−B(n) on the multiplicative complexity of majority.
We show a connection between the deMorgan formula size of a Boolean function and the noise stability of the function. Using this connection, we show that the Fourier spectrum of any balanced Boolean function computed by a deMorgan formula of size s is concentrated on coefficients of degree up to O( √ s). These results have several applications that apply to any function f that can be computed by a deMorgan formula of size s. First, we get that f can be approximated (in L2-norm) with constant error by a polynomial of degree O( √ s). Second, we show an upper bound of O( √ s) on the average sensitivity of f . Our main result stems from a generalization of Khrapchenko’s bound [Khr71], that might be of independent interest, and some Fourier analysis on the Boolean cube. Previous works prove that any function f : {0, 1}n → {0, 1} that can be computed by a deMorgan formula of size s, can be approximated point-wise by a polynomial of degree O(s/2+o(1)) with constant point-wise error. We note that this result can be easily extended to have a polynomial of degree O(t ·s/2+o(1)) that approximates f with point-wise error 2−t, for any t > 0. This was shown in a long line of results in quantum complexity, including [BBC+01] and [FGG08, ACR+07, RS08, Rei09].