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

• URL: http://arxiv.org/abs/2602.13990v1
• Date: Sun, 15 Feb 2026 04:59:48 GMT
• Title: Phase sensitive topological classification of single-qubit measurements in linear cluster states
• Authors: Sougata Bhattacharyya, Sovik Roy,
• Abstract summary: We provide an explicit geometric classification of single-qubit projective measurements on one-dimensional linear cluster states.<n>By establishing an explicit geometrical correspondence between local measurements and topological surgery operations, we represent the cluster state as a linear Hopf chain.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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