Topological Quantum Metrology: Measuring Entanglement Topology via Persistent Homology with Quadratic Scaling

We introduce τ_topo — a topological quantum metric derived from persistent homology of entanglement morphism graphs — that quantifies global entanglement structure invariant under local unitaries and scalable to N-qubit systems with O(N²) measurement complexity. Unlike bipartite entropy, τ_topo captures higher-order connectivity via β₁-barcode length in Rips filtrations of Pauli-weighted state distributions.