Holonomy Obstructions to Composability in Fault-Tolerant Quantum Circuits

This paper identifies a structural gap in fault-tolerant quantum computation between local channel correctness and global composability. Threshold theorems show that if each extended rectangle (ExRec) is locally ε-correct on the code subspace, then long computations can be made reliable. We demonstrate that this local condition, formulated purely at the level of reduced channels, is not by itself sufficient when the physical implementation of ExRecs shares a persistent environment. The key issue is the non-uniqueness of Stinespring dilations: different dilations can produce the same certified channel while differing by nontrivial commutators that act on the logical degrees of freedom. Under coherent sequential composition, these dilation-level effects can accumulate and produce order-one logical deviation after circuit depth on the order of 1/ε, even though each ExRec individually appears arbitrarily accurate. We provide an explicit finite-dimensional construction illustrating this mechanism and identify a sufficient composability condition—trivial logical holonomy (flatness of the dilation bundle)—that restores the standard threshold intuition. The result clarifies an implicit structural assumption underlying composability and separates channel equivalence from dilation equivalence in fault-tolerant reasoning.