Rei Sakuma

Topological data analysis (TDA) extracts global features of data via Betti numbers, but classical computation becomes costly for large simplicial complexes. We propose a scalable quantum TDA algorithm for Betti number estimation based on a block-encoded Dirac operator and logarithmic-depth circuit construction using reconfigurable beam splitter architectures. The method combines shallow Gaussian-Chebyshev spectral filtering with resource-focused complexity analysis. We derive error bounds, estimate gate and qubit requirements, and numerically validate correct Betti number estimation on small instances. Our results indicate that larger instances remain challenging at present but continued progress in quantum hardware may enable increasingly larger problem sizes to become realistic targets in the future.

Quantum topological data analysis is one of the most promising areas for quantum advantage. In particular, the LGZ algorithm is the most basic algorithm for estimating Betti numbers, which are considered to be the essential shape of point clouds. Since LGZ, various quantum algorithms have been proposed, but many of them are based on FTQC and require a large number of quantum resources. In this paper, we propose a QTDA algorithm that does not require ancilla bits using a Hamiltonian based on supersymmetry, which has not been widely used in existing research. In particular, by imposing constraints on the graph, we show that there is always a known 0 eigenvalue and eigenstate in the Hamiltonian, this allows us to use ancilla-free measurement algorithms. Moreover, by using randomization, we reduce the required number of samples.