This talk will explain the structure of Khovanov homology for knots and links and will present a quantum algorithm for its computation that is based on the use of combinatorial Hodge theory. By using combinatorial Hodge theory we can formulate the problem of computing the homology in terms of the determination of the dimension of the eigenvalue one subspace of a unitary operator that is associated with the boundary and coboundary operators of the Khovanov complex. We then use the quantum phase estimation algorithm to find the dimension of this subspace. This is a first foray into the problems of producing a quantum algorithm for the significant knot and link invariant Khovanov homology. There are other deeper problems related to the structure of the chain complex and we shall discuss these difficulties. This paper represents joint work with Sam Lomonaco and Nadya Shirakova.