Quantum lower bound for inverting a permutation with advice

Given a random permutation f : [N ] → [N ] as a black box and y ∈ [N ], we want to output x = f(y). Supplementary to our input, we are given classical advice in the form of a precomputed data structure; this advice can depend on the permutation but not on the input y. Classically, there is a data structure of size Õ(S) and an algorithm that with the help of the data structure, given f(x), can invert f in time Õ(T ), for every choice of parameters S, T , such that S · T ≥ N . We prove a quantum lower bound of T 2 · S ≥ Ω̃(ǫN) for quantum algorithms that invert a random permutation f on an ǫ 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 xj , given the ability to query an N -bit string x at any index except the j-th, and also given m bits of advice that depend on x but not on j.