CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A controlled braid phase is observable

Statement refuted

A scalar phase that is irrelevant for an uncontrolled fixed-sector braid remains irrelevant when that braid is coherently controlled.

Facts & Assumptions

Given: A control qubit with basis $∣ 0 ⟩, ∣ 1 ⟩$ , the target space $C$ , and a control-first controlled-unitary convention.

[F1]

The phase-insensitive projective class identifies $I$ and $- I$ on the one-dimensional target. Projective braid gate and operator metric

[F2]

A controlled global phase becomes a relative control phase and is observable by a two-outcome measurement. A Controlled Global Phase Becomes a Relative Control Phase

Counterexample

Given: The target starts in the unit vector $1 \in C$ , the control starts in $∣ + ⟩ = (∣ 0 ⟩ + ∣ 1 ⟩) / \sqrt{2}$ , and the measurement is in the ${∣ + ⟩, ∣ - ⟩}$ basis.

1.1givenF1algebra

Uncontrolled on $C$ , the gates $I$ and $- I$ differ by the scalar $- 1$ , so they have the same projective class by [F1].

2.1givenF2step 1.1algebra∎

The controlled $I$ is the identity, so it leaves the control in $∣ + ⟩$ and the $+$ outcome has probability $1$ . The controlled $- I$ is $diag (1, - 1)$ on the control; it maps $∣ + ⟩$ to $∣ - ⟩$ , so the same outcome has probability $0$ . Thus the controlled outcome laws differ, refuting the statement.

Depends on

Used by

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Sources

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)