Universal holonomic control of algebras of observables

Abstract

We extend geometric parallel transport from states and degenerate eigenspaces to observable algebras defining generalized quantum subsystems. At fixed block structure, unitary invariance fixes a unique connection, recovering Berry--Wilczek--Zee transport on bands. On positive-dimensional strata, its holonomy is the special-unitary part of the normalizer: all inner logical gates and equal-type sector permutations are attainable. Given a compatible gapped Hamiltonian and access to the required paths, inner gates can be implemented adiabatically by isospectral control of a band refinement.

arXiv comment14 pages, 1 figure; includes appendices and numerical verification code as ancillary files. Revised presentation and clarifications; results unchanged