Manuscript En05-C@1 · En05-C.tex · section · aa213ea6

The theta graph

Stage D: a finite constrained Z2 model › The theta graph · lines 987–1024

En05-C@1#eq:theta-basis, En05-C@1#eq:logical-paulis

Take two vertices joined by three distinct links $1, 2, 3$ . Both vertex constraints are

G = X_{1} X_{2} X_{3} .

The physical Hilbert space has dimension four.

Use the $X$ -eigenbasis with eigenvalues $a, b, c \in {+ 1, - 1}$ . The constraint is $ab c = 1$ . Define an orthonormal physical basis

∣ f, q ⟩_{phys} = ∣ q ⟩_{X, 1} ∣ f q ⟩_{X, 2} ∣ f ⟩_{X, 3}, f, q \in {+ 1, - 1} .

This identifies the vector space with a sector label and a logical qubit. It is not a spatial factorization into the region $S = {1, 2}$ and its complement ${3}$ .

On $H_{phys}$ set

F = X_{3} = X_{1} X_{2}, Q = X_{1}, R = Z_{1} Z_{2} .

Then

F ∣ f, q ⟩ = f ∣ f, q ⟩, Q ∣ f, q ⟩ = q ∣ f, q ⟩, R ∣ f, q ⟩ = ∣ f, - q ⟩ .

Let $e_{f} = (1 + f F) /2$ .

Displayed formulas in this passage

unlabeled · line 991
$G = X_{1} X_{2} X_{3} .$
En05-C@1#eq:theta-basis · line 999
$∣ f, q ⟩_{phys} = ∣ q ⟩_{X, 1} ∣ f q ⟩_{X, 2} ∣ f ⟩_{X, 3}, f, q \in {+ 1, - 1} .$
unlabeled · line 1011
$F = X_{3} = X_{1} X_{2}, Q = X_{1}, R = Z_{1} Z_{2} .$
En05-C@1#eq:logical-paulis · line 1017
$F ∣ f, q ⟩ = f ∣ f, q ⟩, Q ∣ f, q ⟩ = q ∣ f, q ⟩, R ∣ f, q ⟩ = ∣ f, - q ⟩ .$

The theta graph

Displayed formulas in this passage

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