الخيوط الكمومية: كيف تنسج القياسات الروابط سريع

الملخص الأصلي

كما كتبه المؤلفون، بالإنجليزية arXiv:2602.13990

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.