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Standalone and controlled phase accounting for a fixed-sector braid

Statement

For every Fibonacci braid word $b$ with exponent sum $k$ , its physical operator differs from the normalized operator by the common scalar $R_{1}^{k}$ on every fixed-charge block. On the three-anyon SU(2)-lift alphabet $z B_{j}$ , the corresponding scalar is $(R_{1} z^{- 1})^{k}$ . The scalar leaves uncontrolled fixed-sector measurement probabilities unchanged; under a coherent control it is a relative phase. If the scalar is not $1$ , it cannot be discarded without a phase-reference correction; if it is $1$ , the controlled maps coincide and no correction is needed.

Facts & Assumptions

Given: The fixed Fibonacci $F / R$ convention, a finite Artin word $b$ , its exponent sum $k$ , and either an uncontrolled fixed-sector use or a controlled use.

[F1]

The Fibonacci determinant-character supplier proves that exponent sum records the same $R_{1}$ scalar on every charge block and that a controlled scalar becomes relative. The braid phase is a common scalar character

[F2]

An isolated unit-modulus scalar does not change measurement probabilities. Standalone Global Phase Leaves Pure-State POVM Probabilities Unchanged

[F3]

A scalar applied only on the active coherent-control branch can change measurement probabilities. A Controlled Global Phase Becomes a Relative Control Phase

Proof

1.1givenF1algebra

Let $b = σ_{i_{1}}^{ε_{1}} \dots σ_{i_{m}}^{ε_{m}}$ , where $ε_{j} \in {1, - 1}$ . Each generator contributes $R_{1}$ and its inverse contributes $R_{1}^{- 1}$ , so their scalar product is $R_{1}^{\sum_{j} ε_{j}} = R_{1}^{k}$ . The Artin relations preserve exponent sum, hence this factor is independent of the word representative by [F1]. For ${\hat{B}}_{j} = z B_{j}$ , the normalized product contributes $z^{k}$ ; solving for the physical matrix gives $(R_{1} z^{- 1})^{k} \hat{B} (b)$ .

2.1F1F2F3step 1.1∎

On a standalone fixed sector the factor $λ = R_{1}^{k}$ (or $λ = (R_{1} z^{- 1})^{k}$ for the lifted alphabet) has modulus one, so [F2] gives identical outcome probabilities. Under coherent control the block maps are $C (W) = diag (I, W)$ and $C (λ W) = diag (I, λ W)$ . If $λ = 1$ they coincide, so no correction is needed; otherwise [F3] gives a measurement witness distinguishing them. The exact scalar and any correction cost must therefore be recorded as in the phase-reference definition.

Depends on

Used by

A discarded global phase changes a controlled computation Counterexample
Arbitrary-size triple-encoded projective braid approximation Theorem
Fibonacci SU(2) compiler error, length, and bit cost Theorem

Dependency tree · two levels

33 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)