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Three-Ising-anyon braid matrices are Clifford

Example

In the three- $σ$ , total- $σ$ Ising sector, use $F = H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ , $R_{1} = e^{- i π / 8}$ , and $R_{ψ} = e^{3 i π / 8}$ . After removing the common phase $R_{1}$ , the two projective braid generators are $B_{1} = (\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}), B_{2} = H B_{1} H .$

Facts & Assumptions

Given: The displayed fixed Ising fusion sector, its ordered channel basis, and the Pauli matrices $X, Y, Z$ .

[F1]

The Pauli matrices are $X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , $Y = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array})$ , $Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ . The Pauli matrices

[F2]

A unitary is Clifford when it normalizes the phase-aware Pauli group. Clifford normalizer

[F3]

In the three- $σ$ , total- $σ$ sector, the Ising supplier gives $F = H$ , $R_{1} = e^{- i π / 8}$ , and $R_{ψ} = e^{3 i π / 8}$ ; hence $R_{ψ} / R_{1} = i$ . It explicitly limits this to sector data. Ising fusion data and chiral convention

Verification

1.1F3algebra

By [F3], $R_{ψ} / R_{1} = i$ . Removing $R_{1}$ gives $B_{1} = diag (1, i)$ , and recoupling by $H = H^{- 1}$ gives $B_{2} = H B_{1} H$ . Both are unitary because $H$ and $B_{1}$ are unitary. Put $a = (1 + i) / 2$ and $b = (1 - i) / 2$ , so $B_{2} = (\begin{array}{cc} a & b \\ b & a \end{array})$ . Direct multiplication, using $a^{2} + i b^{2} = a$ , $a b (1 + i) = i b$ , and $b^{2} + i a^{2} = - a$ , gives $B_{1} B_{2} B_{1} = (\begin{array}{cc} a & i b \\ i b & - a \end{array}) = B_{2} B_{1} B_{2} .$ Thus these matrices define a unitary representation of the two-generator Artin presentation of $B_{3}$ , independently of any all-label Ising coherence claim.

2.1F1step 1.1algebra

Direct multiplication gives $B_{1} X B_{1}^{*} = Y$ , $B_{1} Y B_{1}^{*} = - X$ , and $B_{1} Z B_{1}^{*} = Z$ ; also $B_{2} X B_{2}^{*} = X$ , $B_{2} Y B_{2}^{*} = Z$ , and $B_{2} Z B_{2}^{*} = - Y$ . Thus each generator conjugates every Pauli matrix to a signed Pauli matrix.

3.1F2step 1.1step 2.1algebra∎

Products and inverses of Pauli normalizers again normalize the Pauli group: for normalizers $A, B$ , $(A B) P (A B)^{*} = A (B P B^{*}) A^{*} = P$ , and the inverse property follows by conjugating the equality $A P A^{*} = P$ by $A^{*}$ . The scalar phase removed in step 1.1 has trivial conjugation action, so the physical braids are projectively Clifford in this sector.

Remarks

This is a projective statement for an isolated fixed-charge sector. The scalar phase cannot be discarded when the braid is coherently controlled. The supplier gives sector data only; this item makes no all-label Ising category claim.

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)
John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)