Quantum algorithms achieve polylogarithmic complexity for Betti number estimation and homology testing via block-encoded Laplacians and cohomological projections, claiming exponential speedups under sparsity assumptions.
A quantum algorithm for khovanov homology.arXiv preprint arXiv:2501.12378
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Non-Hermitian multi-band twister models are simulated on quantum hardware via a direct measurement protocol that extracts braid information and knot invariants such as Alexander and Jones polynomials, demonstrated on the Hopf chain and Solomon's knot.
Pro-tangles are introduced as Boolean-cube functors and used to build a representable Khovanov simplicial presheaf whose Boolean-cube spectral sequence converges to total homology with explicit E1 page from reduced tangles.
citing papers explorer
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New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes
Quantum algorithms achieve polylogarithmic complexity for Betti number estimation and homology testing via block-encoded Laplacians and cohomological projections, claiming exponential speedups under sparsity assumptions.
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Digital Simulation of Non-Hermitian Knotted Bands on Quantum Hardware
Non-Hermitian multi-band twister models are simulated on quantum hardware via a direct measurement protocol that extracts braid information and knot invariants such as Alexander and Jones polynomials, demonstrated on the Hopf chain and Solomon's knot.
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Khovanov homology: pro-tangles, derived colimits and spectral sequences
Pro-tangles are introduced as Boolean-cube functors and used to build a representable Khovanov simplicial presheaf whose Boolean-cube spectral sequence converges to total homology with explicit E1 page from reduced tangles.