We prove three theorems on controlled-unitary gate topology in quantum circuits. Theorem 1: star and chain CNOT circuits produce identical outputs if and only if the middleman qubit is initialized to |0⟩, failing for 100% of random initial conditions. Theorem 2: the fidelity between star and chain outputs follows F = cos⁴(α), shown independent of the opening gate for all G ∈ U(2), establishing non-transitivity as a topological rather than quantum-superposition effect. Theorem 3: star topologies provide exact fault isolation — arbitrary unitary errors on the middleman leave the hub-endpoint reduced state invariant — while chain topologies propagate such errors. All results are experimentally verified on IBM Torino quantum hardware.
Mattson (Fri,) studied this question.