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This paper argues that standard Berry-phase, Chern-number, and Wilson-loop diagnostics can't tell whether a non-Abelian holonomy acts as a local two-qubit operation or as an entangling gate. The work is theoretical and uses topological boundary-state models (BHZ, SSH, BBH) as testbeds, so its near-term relevance to quantum hardware is unproven. Readers interested in geometric gates should note that the criterion, whether the holonomy stays inside an extracted local subgroup, is a concrete diagnostic they could check on their own models. The excerpt is truncated before the main results, so the claims can't be fully assessed yet.
Loop-dependent entangling holonomies in localized topological quartets
Non-Abelian adiabatic holonomies are used both to diagnose topology and to implement geometric quantum operations. Yet the standard descriptors of such transport --- Berry phases, Chern numbers, determinant phases, and Wilson-loop spectra --- do not determine whether the resulting holonomy acts as local logical operations on an extracted subsystem structure. We identify this missing information [...]
An isolated four-state manifold can admit a pointwise two-qubit description and still return from a closed loop by a nonlocal gate. The same quartet may be almost local on one loop and entangling on another. In the BHZ ribbon studied below, co-rotating and counter-rotating edge-field loops act on the same helical quartet, but only the counter-rotating loop leaves the extracted local subgroup by an order-one amount. Standard reduced Berry data do not resolve this difference. Chern numbers, determinant phases, and Wilson-loop eigenphases record phases or spectra of the loop operator. The question asked here is different. In the extracted local frame, can the microscopic holonomy be written as two independent single-qubit actions? Nearly identical eigenphase data can hide different gate classes, and a large Berry phase can occur on an almost local loop. What matters is whether the holonomy stays inside the extracted local subgroup. The three benchmarks used below are standard settings in which localized four-state manifolds arise. The Bernevig-Hughes-Zhang (BHZ) model describes a two-dimensional quantum spin Hall insulator and supplies a helical pair on each open boundary [1]. The spinful Su-Schrieffer-Heeger (SSH) chain is a one-dimensional dimerized system whose two end modes are doubled by spin [2]. The…
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