Efficient quantum algorithm for approximating Khovanov homology

Design a quantum algorithm that, given a knot diagram with m crossings, runs in time polynomial in m and outputs additive approximations to the Betti numbers of Khovanov homology Kh^{i,j}(K) (or an equivalent efficient approximation of Khovanov homology).

Background

Khovanov homology is a categorification of the Jones polynomial and is strictly stronger as a knot invariant. While efficient quantum algorithms exist for approximating the Jones polynomial, no general efficient quantum algorithm is known for Khovanov homology. The paper proposes a quantum approach based on the Hodge Laplacian and a pre-thermalization procedure, but its overall efficiency depends on spectral gaps and thermalization conditions that are not established in general.