Quantum algorithms and approximating polynomials for composed functions with shared inputs

Abstract

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m-bit Boolean function and consider an n-bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f), we give an algorithm for evaluating F using O~(Q(f)⋅n) quantum queries. This improves on the bound of O(Q(f)⋅n) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f.By recursively applying our composition theorems, we obtain a nearly optimal O~(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0 circuits.As an additional consequence, we show that AC0∘⊕ circuits of depth d+1 require size Ω~(n1/(1−2−d))≥ω(n1+2−d) to compute the Inner Product function even on average. The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).