Phase sensitive topological classification of single-qubit measurements in linear cluster states

We provide an explicit geometric classification of single-qubit projective measurements on one-dimensional linear cluster states within a topological framework. By establishing an explicit geometrical correspondence between local measurements and topological surgery operations on an associated link model i.e. a measurement surgery correspondence, we represent the cluster state as a linear Hopf chain. Within this model, measurements in the computational (Z) basis act as topological severance in case of bulk measurements while boundary pruning happens for end measurements of qubits. In contrast, transverse (X) basis measurements remove the measured qubit while splicing its neighbours, preserving connectivity through real valued correlations. We show that lateral (Y) basis measurements also preserve connectivity but generate intrinsically complex phase factors that are not captured by unframed link models, rendering X and Y measurements topologically indistinguishable at the level of connectivity alone. To resolve this ambiguity, we introduce a framed ribbon representation in which quantum phases are encoded as geometric twists, with chiral pm 90^circ} twists corresponding to the phases\pm i$. This framing yields a phase-sensitive and outcome resolved topological description of all single qubit measurements on linear cluster states. Our approach provides a unified geometric interpretation of measurement-induced entanglement transformations in measurement-based quantum computation, revealing that quantum phases correspond directly to framed topological invariants. The work is restricted to one-dimensional linear cluster states and single-qubit measurements in the Pauli bases.