Efficient quantum algorithm for identifying hidden polynomials

Abstract

We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with $d$ elements. The hidden functions of the generalized problem are not restricted to be linear but can also be $m$-variate polynomial functions of total degree $n\geq 2$. The problem of identifying hidden $m$-variate polynomials of degree less or equal to $n$ for fixed $n$ and $m$ is hard on a classical computer since $\Omega(\sqrt{d})$ black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of $d$ with constant probability and that has a running time that is only polylogarithmic in $d$.

Hidden Symmetry Subgroup Problems

“…Since E d is finite it follows that for fixed d we can solve HPGP(F q , 1, d) in quantum polynomial time for all finite fields. Together with Fact 4 (a) this improves the overall result of [9]: the HPGP(F q , n, d) can be solved efficiently for all finite fields when n and d are constant. We further improve this result in the next section where we present a more powerful reduction of the multivariate problem to the univariate case.…”

“…These groups turn out to have a base of size d, and therefore our general reduction to the related HSP applies (Theorem 4). Based on this reduction, we improve the results of [9] by showing that there is a quantum polynomial time algorithm for the HPGP over every field when the degree of the polynomials is constant (Theorem 5).…”

“…We are not aware of any results regarding the quantum computational complexity even in the univariate quadratic case. For the HPGP, Decker, Draisma and Wocjan [9] showed the following.…”

“…These reductions explain why it was possible to construct the algorithm of [9] in close analogy with the pretty good measurement framework of [3] for semidirect product groups.…”

“…In [9], Decker, Draisma and Wocjan defined a related problem that we refer to as the hidden polynomial graph problem (HPGP) to distinguish it from the HPP. Here, similarly to the HPP, the hidden object is a polynomial, but the oracle is more powerful because it can also be queried on the graphs that are defined by the polynomial functions.…”