Quantum computation of zeta functions of curves

Abstract.

We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field $\mathbb{F}_{q}$, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of a curve, plus a recipe for recovering a Weil polynomial from enough of its cyclic resultants. The latter effectivizes a result of Fried in a restricted setting.