Khovanov Homology of Wilson Loops in Topologically Protected Quantum Memories

We present a rigorous categorification of the expectation values of Wilson loops within (2 + 1)-dimensional topological quantum field theories (TQFTs), specifically targeting the stability conditions of topologically protected quantum memories. By lifting the Jones polynomial of the Wilson loop WγW_\gammaWγ to a bigraded Khovanov homology complex HKhi,j(Wγ)H^{i,j}_{\text{Kh}}(W_\gamma)HKhi,j(Wγ), we construct a spectral sequence Erp,qE^{p,q}_rErp,q converging to the symplectic Floer homology of the underlying Lagrangian submanifold in the moduli space of flat connections. We demonstrate that the graded Euler characteristic of this complex recovers the quantum dimension of the anyonic excitation associated with γ\gammaγ. Furthermore, we prove that the preservation of the homological grading under local Hamiltonian perturbations provides a robust invariant for the fault tolerance of the memory manifold. This framework utilizes derived categories of matrix factorizations and motivic cohomology to establish a trans-finite bound on the decoherence rates of logical qubits encoded in the ground state degeneracy.