Fourier Growth of Parity Decision Trees

TL;DR

This paper establishes tight bounds on the Fourier spectrum of parity decision trees, extending previous results for standard decision trees, and applies these bounds to analyze the complexity of the Forrelation problem.

Contribution

It provides nearly tight Fourier bounds for parity decision trees and introduces a new proof approach using random walks, simplifying previous methods.

Findings

Fourier coefficients at level ll are bounded by d^{ll/2} (ll ( log(n)))^ll.

The k-fold Forrelation problem has ( ^{1-1/k}) randomized parity decision tree complexity.

A similar Fourier bound is proved for noisy decision trees of bounded cost.

Abstract

We prove that for every parity decision tree of depth $d$ on $n$ variables, the sum of absolute values of Fourier coefficients at level $\ell$ is at most $d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$. Our result is nearly tight for small values of $\ell$ and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the $k$-fold Forrelation problem has (randomized) parity decision tree complexity $\tilde{\Omega}\left(n^{1-1/k}\right)$, while having quantum query complexity $\lceil k/2\rceil$. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-$\ell$ Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-$\ell$ walks can be computed by the intermediate values of level $\le \ell-1$ walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive. In addition, we prove a similar bound for noisy decision trees of cost at most $d$ -- a model that was recently introduced by Ben-David and Blais (FOCS, 2020).