Recently the celebrated Khovanov Homology was introduced as a target for Topological Quantum Computation given that the Khovanov Homology provides a generalization of the Jones polynomal and then it is possible to think about of a generalization of the Aharonov.-Jones-Landau algorithm. Recently, Lipshitz and Sarkar introduced a space-level refinement of Khovanov homology. which is called Khovanov Homotopy. This refinement induces a Steenrod square operation Sq² on Khovanov homology which they describe explicitly and then some computations of Sq² were presented. Particularly, examples of links with identical integral Khovanov homology but with distinct Khovanov homotopy types were showed. In the presente work we will introduce possible quantum algorithms for the Lipshitz- Sarkar-Steenrod square for Khovanov Homolog and their possible simulations using computer algebra.

Date Published: 28 May 2013
PDF: 13 pages
Proc. SPIE 8749, Quantum Information and Computation XI, 87490K (28 May 2013); doi: 10.1117/12.2016298

Published in SPIE Proceedings Vol. 8749:
Quantum Information and Computation XI
Eric Donkor; Andrew R. Pirich; Howard E. Brandt, Editor(s)