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

TL;DR

This work provides a phase-sensitive geometric framework for understanding single-qubit measurements on 1D linear cluster states by mapping measurement operations to topological surgeries on a Linear Hopf Chain. It introduces a measurement-cutting correspondence and a framed ribbon model, where Z-basis cuts sever connectivity, X-basis splices preserve connectivity with real correlations, and Y-basis splices preserve connectivity but imprint chiral ±i phases via ribbon twists. The key contribution is resolving the X versus Y ambiguity inherent in unframed link topologies by encoding quantum phases as discrete twists, enabling a complete phase-sensitive topological classification of all single-qubit measurements on LCS. The framework unifies measurement-induced entanglement transformations in MBQC, offers intuitive visualization of by-product operators, and opens avenues for extension to higher dimensions, dynamic circuits, fault-tolerant schemes, and non-stabilizer resources, with potential impact on geometric approaches to quantum computation.

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.

Figures (10)

• Figure 1: The Hopf link
• Figure 2: The Linear Hopf Chain Model.(a) 2D diagrammatic representation of the 1D cluster state connectivity. (b) 3D visualization of the same state as a series of physically interlinked rings. In this dual picture, quantum measurements correspond to topological operations (cutting or splicing) performed on these links.
• Figure 3: Topological Cutting in the Z-Basis - Internal Severance. Measuring a bulk qubit ($Q_k$) physically removes the central ring. Unlike Splicing (which fuses neighbours), Severance destroys the path, leaving two unlinked, independent segments ($R=1$).
• Figure 4: Topological Cutting in the $Z$-Basis - Boundary Pruning. Measuring an end qubit ($Q_1$) removes the terminal ring. The chain remains connected but shortens ($N \to N-1$). Contrast this with X-basis propagation, shown in Fig. \ref{['fig:x_basis_end']}: here, the boundary moves only one step, to $Q_2$.
• Figure 5: Topological Splicing in X-Basis - Internal Splicing. The critical contrast to Z-severance. Measuring $Q_k$ fuses the neighbors $Q_{k-1}$ and $Q_{k+1}$ into a direct link, preserving the linear topology ($R=2$) instead of breaking it.