Topological signatures of quantum criticality from fidelity-based persistence

We develop a rigorous mathematical framework for the classification of quantum phases and the detection of quantum phase transitions using persistent homology and topological data analysis (TDA). Beginning from the quantum fidelity metric on a parameterised family of ground states, we construct a filtered simplicial complex — the quantum Vietoris–Rips complex — whose persistent homology groups encode the topological structure of the ground state manifold. We prove that quantum phase transitions correspond to features of infinite persistence in the thermodynamic limit, and establish a finite-size scaling theory that yields a new critical exponent α=(2−η)/2ν−d/2. Stability of the persistence diagram under Hamiltonian perturbations is proved via a bottleneck theorem adapted to the fidelity metric. A sheaf-theoretic reformulation identifies topological phases with cohomological obstructions, and exact calculations for the transverse-field Ising model give α=3/8. A complete photonic experimental protocol for measuring α on a programmable 50–100 mode linear optical network closes the paper.