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

TL;DR

This paper introduces a topological framework for classifying single-qubit measurements on linear cluster states, linking measurement effects to topological operations and phases via a geometric and framed ribbon model.

Contribution

It provides a novel phase-sensitive topological classification of measurements on linear cluster states using a framed link model that encodes quantum phases as geometric twists.

Findings

Z measurements cause topological severance or boundary pruning.

X and Y measurements preserve connectivity but differ in phase effects.

Framed ribbon model encodes quantum phases as geometric twists.

Abstract

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.