Quantum Spectral Algorithms for Khovanov Cohomology via Laplacian Formalism and Frobenius Cobordisms

Abstract

We present a comprehensive quantum algorithm for computing Khovanov cohomology, a categorification of the Jones polynomial, together with a clear narrative that connects its motivation, construction, and implementation. By simulating Laplacian operators derived from the Khovanov cochain complex, we estimate cohomological dimensions using quantum phase estimation, thermal sampling, and Quantum Singular Value Transformation (QSVT). The framework explicitly exploits the Frobenius algebra structure and cobordism maps, showing how these topological tools translate naturally into unitary quantum operations. Worked examples for the trefoil knot and Hopf link illustrate each step, from encoding the cube of resolutions to constructing differentials and Laplacians. This narrative approach not only refines and operationalizes the Laplacian method introduced by previous works but also provides a practical roadmap for extending categorified invariants into the realm of quantum algorithms suitable for near-term devices.