Phase sensitive topological classification of single-qubit measurements in linear cluster states
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| Format: | Preprint |
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2026
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| _version_ | 1866912906154606592 |
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| author | Bhattacharyya, Sougata Roy, Sovik |
| author_facet | Bhattacharyya, Sougata Roy, Sovik |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_13990 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Phase sensitive topological classification of single-qubit measurements in linear cluster states Bhattacharyya, Sougata Roy, Sovik Quantum Physics 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. |
| title | Phase sensitive topological classification of single-qubit measurements in linear cluster states |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2602.13990 |