Formula lower bounds via the quantum method

A de Morgan formula over Boolean variables x1,…,xn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires a large formula is a central open question in computational complexity. While we believe that some explicit functions require exponential formula size, currently the best lower bound for an explicit function is the Ω(n3) lower bound for Andreev's function. A long line of work in quantum query complexity, culminating in the work of Reichardt [SODA, 2011], proved that for any formula of size s, there exists a polynomial of degree at most O(√s) that approximates the formula up to a small point-wise error. This is a classical theorem, arguing about polynomials and formulae, however the only known proof for it involves quantum algorithms. We apply Reichardt result to obtain the following: (1) We show how to trade average-case hardness in exchange for size. More precisely, we show that if a function f cannot be computed correctly on more than 1/2 + 2-k of the inputs by any formula of size at most s, then computing f exactly requires formula size at least Ω(k) · s. As an application, we improve the state of the art formula size lower bounds for explicit functions by a factor of Ω(logn). (2) We prove that the bipartite formula size of the Inner-Product function is Ω(n2). (A bipartite formula on Boolean variables x1,…,xn and y1, …, yn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves can compute any function of either the x or y variables.) We show that any bipartite formula for the Inner-Product modulo 2 function, namely IP(x,y) = Σi=1n xi yi (mod 2), must be of size Ω(n2), which is tight up to logarithmic factors. To the best of our knowledge, this is the first super-linear lower bound on the bipartite formula complexity of any explicit function.