Quantum lower bound for inverting a permutation with advice (pp0901-0913)
          Aran Nayebi, Scott Aaronson, Aleksandrs Belovs, Luca Trevisan
          doi: https://doi.org/10.26421/QIC15.11-12-1
Abstracts: Given a random permutation f : [N] → [N] as a black box and y ∈ [N], we want to output x = f−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 O˜(S) and an algorithm that with the help of the data structure, given f(x), can invert f in time O˜(T), for every choice of parameters S, T, such that S · T ≥ N. We prove a quantum lower bound of T 2 · S = 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(p N/m) quantum lower bound for the simpler but related Yao’s box problem, which is the problem of recovering a bit xj , 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.
Key words: quantum lower bound, one-way function, random permutation, time-space tradeof