Abstract
We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Letfbe anm-bit Boolean function and consider ann-bit functionFobtained by applyingfto conjunctions of possibly overlapping subsets ofnvariables. Iffhas quantum query complexityQ(f), we give an algorithm for evaluatingFusingO~(Q(f)⋅n)quantum queries. This improves on the bound ofO(Q(f)⋅n)that follows by treating each conjunction independently, and our bound is tight for worst-case choices off. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree off.By recursively applying our composition theorems, we obtain a nearly optimalO~(n1−2−d)upper bound on the quantum query complexity and approximate degree of linear-size depth-dAC0circuits. 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 AC0circuits.As an additional consequence, we show that AC0∘⊕circuits of depthd+1require sizeΩ~(n1/(1−2−d))≥ω(n1+2−d)to compute the Inner Product function even on average. The previous best size lower bound wasΩ(n1+4−(d+1))and only held in the worst case (Cheraghchi et al., JCSS 2018).
Highlights
In the query, or black-box, model of computation, an algorithm aims to evaluate a known Boolean function f : {0, 1}n → {0, 1} on an unknown input x ∈ {0, 1}n by reading as few bits of x as possible
Our main result shows that a shared-input composition between a function f and the AND function always has substantially lower quantum query complexity than the block composition f ◦√ANDn
We complement our result on the quantum query complexity of shared-input compositions with an analogous result for approximate degree
Summary
Black-box, model of computation, an algorithm aims to evaluate a known Boolean function f : {0, 1}n → {0, 1} on an unknown input x ∈ {0, 1}n by reading as few bits of x as possible. One of the most basic questions one can ask about query complexity, or any complexity measure of Boolean functions, is how it behaves under composition. Given functions f and g, and a method of combining these functions to produce a new function h, how does the query complexity of h depend on the complexities of the constituent functions f and g?. The simplest method for combining functions is block composition, where the inputs to f are obtained by applying the function g to independent sets of variables. The query complexity of f ◦ g is at most the product of the complexities of f and g.1
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