Sensitivity, Affine Transforms and Quantum Communication Complexity

Abstract

In this paper, we study the Boolean function parameters sensitivity (${\mathsf {s}}$), block sensitivity (${\mathsf {bs}}$), and alternation (${\mathsf {alt}}$) under specially designed affine transforms and show several applications. For a function $f:{\mathbb {F}}_2^n \rightarrow \{0,1\}$, and $A = Mx+b$ for $M \in {\mathbb {F}}_2^{n \times n}$ and $b \in {\mathbb {F}}_2^n$, the result of the transformation g is defined as $\forall x \in {\mathbb {F}}_2^n, g(x) = f(Mx+b)$.

As a warm up, we study alternation under linear shifts (when M is restricted to be the identity matrix) called the shift invariant alternation (the smallest alternation that can be achieved for the Boolean function f by shifts, denoted by ${\mathsf {salt}}(f)$). By a result of Lin and Zhang [12], it follows that ${\mathsf {bs}}(f) \le O({\mathsf {salt}}(f)^2{\mathsf {s}}(f))$. Thus, to settle the Sensitivity Conjecture ($\forall ~f, {\mathsf {bs}}(f) \le {\mathsf {poly}}({\mathsf {s}}(f))$), it suffices to argue that $\forall ~f, {\mathsf {salt}}(f) \le {\mathsf {poly}}({\mathsf {s}}(f))$. However, we exhibit an explicit family of Boolean functions for which ${\mathsf {salt}}(f)$ is $2^{\varOmega ({\mathsf {s}}(f))}$.

Going further, we use an affine transform A, such that the corresponding function g satisfies ${\mathsf {bs}}(f,0^n) \le {\mathsf {s}}(g)$, to prove that for $F(x,y) {\mathop {=}\limits ^{\mathrm{def}}}f(x \wedge y)$, the bounded error quantum communication complexity of F with prior entanglement, $Q^*_{1/3}(F)$ is $\varOmega (\sqrt{{\mathsf {bs}}(f,0^n)})$. Our proof builds on ideas from Sherstov [17] where we use specific properties of the above affine transformation. Using this, we show the following.

• (a) For a fixed prime p and an $\epsilon$, $0< \epsilon < 1$, any Boolean function f that depends on all its inputs with ${\mathsf {deg}}_p(f) \le (1-\epsilon )\log n$ must satisfy $Q^*_{1/3}(F) = \varOmega \left( \frac{n^{\epsilon /2}}{\log n} \right)$. Here, ${\mathsf {deg}}_p(f)$ denotes the degree of the multilinear polynomial over ${\mathbb {F}}_p$ which agrees with f on Boolean inputs.

• (b) For Boolean function f such that there exists primes p and q with ${\mathsf {deg}}_q(f) \ge \varOmega ({\mathsf {deg}}_p(f)^\delta )$ for $\delta > 2$, the deterministic communication complexity - ${\mathsf {D}}(F)$ and $Q^*_{1/3}(F)$ are polynomially related. In particular, this holds when ${\mathsf {deg}}_p(f) = O(1)$. Thus, for this class of functions, this answers an open question (see [2]) about the relation between the two measures.

Restricting back to the linear setting, we construct linear transformation A, such that the corresponding function g satisfies, ${\mathsf {alt}}(f) \le 2{\mathsf {s}}(g)+1$. Using this new relation, we exhibit Boolean functions f (other than the parity function) such that ${\mathsf {s}}(f)$ is $\varOmega (\sqrt{\mathsf {sparsity}(f)})$ where $\mathsf {sparsity}(f)$ is the number of non-zero coefficients in the Fourier representation of f.