Efficient Quantum Algorithm for the Hidden Parabola Problem

We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of the abelian Hidden Subgroup Problem (HSP) over groups of the form Zp × Zp, where the hidden subgroups are generated by (1, a) for different a ∈ Zp. These subgroups and their cosets correspond to graphs of linear functions on Zp. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be polynomial functions of degree n ≥ 2. Analogously to the HSP, for a fixed degree n the Hidden Polynomial Function Graph Problem is hard on a classical computer as its query complexity is polynomial in p. To solve this problem on a quantum computer in time polylogarithmic in p, we first reduce it to a quantum state identification problem and then we use the pretty good measurement (PGM) approach to construct measurements for distinguishing the states. We relate the success probability and implementation of the PGM to a certain classical problem involving polynomial equations. We present an efficient quantum algorithm for the case of hidden parabola by establishing that the success probability of the PGM is lower bounded by a constant and that the PGM can be implemented efficiently.