Speaker
Description
Topological data analysis (TDA) extracts robust global features from high-dimensional data by representing data sets as simplicial complexes. A central challenge is to compute topological invariants of these complexes, especially the $k^{\mathrm{th}}$ Betti number, whose nonzero value indicates the presence of a $k$-dimensional hole. In this work, we study the decision problem of determining whether the $k^{\mathrm{th}}$ Betti number is nonzero for the clique complex of a graph $G$. Using a hard-core fermionic construction, we encode this problem into the ground-space structure of an associated fermionic Hamiltonian. Betti-number detection is thereby reduced to a ground-state problem in the $(k+1)$-fermion sector. To search for low-energy states of this Hamiltonian, we introduce a progressive multi-stage optimization protocol based on variational quantum circuits. The method is benchmarked on island-graph families with analytically controlled Betti numbers and spectral gaps. Finally, we derive resource estimates for Erdős--Rényi random graphs in the regime where the associated clique complexes have nontrivial topology.
| Topic | Quantum variational circuits and their applications |
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