Quantum Lower Bound for Inverting a Permutation with Advice

Abstract

Given a random permutation f : [N] → [N] as a black box and y ∈ [N], we want to output x = f[superscript −1](y). Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but not on the input y. Classically, there is a data structure of size [~ over O](S) and an algorithm that with the help of the data structure, given f(x), can invert f in time [~ over O](T), for every choice of parameters S, T, such that S · T ≥ N. We prove a quantum lower bound of T[superscript 2] · S = [~ over Ω](εN) for quantum algorithms that invert a random permutation f on an ε fraction of inputs, where T is the number of queries to f and S is the amount of advice. This answers an open question of De et al. We also give a Ω(√N/m) quantum lower bound for the simpler but related Yao’s box problem, which is the problem of recovering a bit x[subscript j], given the ability to query an N-bit string x at any index except the j-th, and also given m bits of classical advice that depend on x but not on j.