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A discarded global phase changes a controlled computation

Statement

On one target qubit, $I_{2}$ and $- I_{2}$ differ only by a global phase and induce the same standalone ray and outcome probabilities. Controlled access changes the comparison. With control first, $C (I_{2}) = I_{4}, C (- I_{2}) = (\begin{array}{cc} I_{2} & 0 \\ 0 & - I_{2} \end{array}) = Z \otimes I_{2} .$ On input $∣ + ⟩ ∣ 0 ⟩$ , these give $∣ + ⟩ ∣ 0 ⟩$ and $∣ - ⟩ ∣ 0 ⟩$ . Measuring the control in the $X$ basis is respectively deterministic for $+$ and $-$ , so the total-variation distance is one. Thus projective equivalence of an isolated braid gate is not interchangeable with equality under coherent control.

Diagram

The diagram uses a filled control dot on the upper wire and the scalar phase $- I$ on the lower target branch. The final $X$ measurement reads the relative phase on the control. It is a circuit convention only; the matrix calculation below proves the claim.

Facts & Assumptions

Given: A single control qubit, a target qubit, and the two phase-related target gates $I_{2}$ and $- I_{2}$ .

[F1]

Standalone global phase does not change ray probabilities, while a controlled global phase becomes relative. Standalone and controlled phase accounting for a fixed-sector braid

[F2]

The controlled-phase supplier states the control-target relative-phase effect. A Controlled Global Phase Becomes a Relative Control Phase

Proof

1.1F1givenalgebra

Both $I_{2}$ and $- I_{2}$ send every target vector to the same ray, since the outputs differ by scalar $- 1$ ; hence their standalone outcome probabilities coincide by [F1]. In the ordered control basis $∣ 0 ⟩, ∣ 1 ⟩$ , the controlled block matrix is $diag (I_{2}, - I_{2}) = Z \otimes I_{2}$ .

2.1F2step 1.1algebra∎

Applying this matrix to $∣ + ⟩ ∣ 0 ⟩ = (∣ 0 ⟩ + ∣ 1 ⟩) ∣ 0 ⟩ / \sqrt{2}$ gives $∣ - ⟩ ∣ 0 ⟩$ , whereas controlled $I_{2}$ leaves the input unchanged. The $X$ -basis outcome laws are point masses on opposite outcomes, so their total-variation distance is $1$ . This is the relative-phase effect in [F2].

Depends on

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Sources

John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)
Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)