Sensitivity, Affine Transforms and Quantum Communication Complexity

TL;DR

This paper explores how affine transformations affect Boolean function parameters like sensitivity and block sensitivity, and applies these insights to quantum communication complexity, revealing new relationships and limitations in complexity measures.

Contribution

It introduces new affine transform techniques to relate sensitivity, block sensitivity, and alternation, impacting quantum communication complexity and addressing open problems.

Findings

Existence of functions with exponential shift-invariant alternation relative to sensitivity

Affine transformations can bound block sensitivity by sensitivity in certain functions

Results challenge potential approaches to the XOR Log-Rank conjecture via Sensitivity conjecture

Abstract

$\newcommand{\F}{\mathbb{F}}$We study the Boolean function parameters sensitivity ($s$), block sensitivity ($bs$), and alternation ($alt$) under specially designed affine transforms. For a function $f:\F_2^n\to \{0,1\}$, and $A=Mx+b$ for $M \in \F_2^{n\times n}$ and $b\in \F_2^n$, the result of the transformation $g$ is defined as $\forall x\in\F_2^n, g(x)=f(Mx+b)$. We study alternation under linear shifts ($M$ is the identity matrix) called the shift invariant alternation (denoted by $salt(f)$). We exhibit an explicit family of functions for which $salt(f)$ is $2^{\Omega(s(f))}$. We show an affine transform $A$, such that the corresponding function $g$ satisfies $bs(f,0^n) \le s(g)$, using which we proving that for $F(x,y)=f(x\land y)$, the bounded error quantum communication complexity of $F$ with prior entanglement, $Q^*_{1/3}(F)=\Omega(\sqrt{bs(f,0^n)})$. Our proof builds on ideas from Sherstov (2010) where we use specific properties of the above affine transformation. We show, * For a prime $p$ and $0<\epsilon<1$, any $f$ with $deg_p(f)\le(1-\epsilon)\log n$ must satisfy $Q^*_{1/3}(F) = \Omega(\frac{n^{\epsilon/2}}{\log n})$. Here, $deg_p(f)$ denotes the degree of the multilinear polynomial of $f$ over $\F_p$. * For any $f$ such that there exists primes $p$ and $q$ with $deg_q(f) \ge \Omega(deg_p(f)^\delta)$ for $\delta > 2$, the deterministic communication complexity - $D(F)$ and $Q^*_{1/3}(F)$ are polynomially related. In particular, this holds when $deg_p(f) = O(1)$. Thus, for this class of functions, this answers an open question (see Buhrman and deWolf (2001)) about the relation between the two measures. We construct linear transformation $A$, such that $g$ satisfies, $alt(f) \le 2s(g)+1$. Using this, we exhibit a family of Boolean functions that rule out a potential approach to settle the XOR Log-Rank conjecture via a proof of Sensitivity conjecture [Hao Huang (2019)].