Sensitivity, Affine Transforms and Quantum Communication Complexity
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We study the Boolean function parameters sensitivity (s), block sensitivity (bs), and alternation (alt) under specially designed affine transforms. For a function f:_2^n →{0,1}, and A = Mx+b for M ∈_2^n × n and b ∈_2^n, the result of the transformation g is defined as ∀ x ∈_2^n, g(x) = f(Mx+b). We study alternation under linear shifts (when M is the identity matrix) called the shift invariant alternation (denoted by salt(f)). By a result of Lin and Zhang (2017), it follows that bs(f) < O(salt(f)^2s(f)). Thus, to settle the Sensitivity Conjecture, it suffices to argue that ∀ f, salt(f) < poly(s(f)). However, we exhibit an explicit family of functions for which salt(f) is 2^Ω(s(f)). We show an affine transform A, such that the corresponding function g satisfies bs(f,0^n) < s(g), using which we proving that for F(x,y) := f(x y), the bounded error quantum communication complexity of F with prior entanglement, Q^*_1/3(F) = Ω(√(bs(f,0^n))). Our proof builds on ideas from Sherstov (2010) where we use specific properties of the above affine transformation. Using this, we show, * For a fixed prime p and 0 < ϵ < 1, any f with deg_p(f) < (1-ϵ) n must satisfy Q^*_1/3(F) = Ω (n^ϵ/2/ n ). Here, deg_p(f) denotes the degree of the multilinear polynomial of f over _p . * For f such that there exists primes p and q with deg_q(f) >Ω(deg_p(f)^δ) for δ > 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.
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