Uploaded March 2026 | Updated September 2026, 3 weeks ago
In this talk, we present a quantum algorithm developed in collaboration with Alexander Schmidhuber, Michele Reilly, Paolo Zanardi, and Seth Lloyd for estimating the ranks of Khovanov homology groups utilizing a combinatorial Hodge theory. We introduce the notion of “harmonic Khovanov homology,” where we identify unique representatives of homology classes as the kernel of a Khovanov-Hodge Laplacian. While classical computation of these groups scales exponentially with the number of crossings, our quantum algorithm provides an efficient alternative, provided the Laplacian satisfies certain spectral conditions. We will discuss joint work with Jernej Grlj exploring the “higher spectrum” of this Laplacian and the question of what information the non-zero eigenvalues encode about the underlying link diagram. We conclude with numerical evidence for the algorithm’s efficiency and open questions regarding analytic bounds on the spectral gap for general knots.
For more information please visit: simonsfoundation.org/event/simons-collaboration-on-new-structures-in-low-dimensional-topology-annual-meeting-2026
In this talk, we present a quantum algorithm developed in collaboration with Alexander Schmidhuber, Michele Reilly, Paolo Zanardi, and Seth Lloyd for estimating the ranks of Khovanov homology groups utilizing a combinatorial Hodge theory. We introduce the notion of “harmonic Khovanov homology,” where we identify unique representatives of homology classes as the kernel of a Khovanov-Hodge Laplacian. While classical computation of these groups scales exponentially with the number of crossings, our quantum algorithm provides an efficient alternative, provided the Laplacian satisfies certain spectral conditions. We will discuss joint work with Jernej Grlj exploring the “higher spectrum” of this Laplacian and the question of what information the non-zero eigenvalues encode about the underlying link diagram. We conclude with numerical evidence for the algorithm’s efficiency and open questions regarding analytic bounds on the spectral gap for general knots.
For more information please visit: simonsfoundation.org/event/simons-collaboration-on-new-structures-in-low-dimensional-topology-annual-meeting-2026
