Manuscript En02-B@1 · En02-B-v1.tex · section · ca28dee4

The exact five-qubit perfect-code building block

The exact five-qubit perfect-code building block · lines 543–611

En02-B@1#eq:stabilizers, En02-B@1#eq:encoding

The five-qubit code is a standard quantum-error-correction benchmark Laflamme. We use the stabilizer convention in the perfect-tensor construction HaPPY (Appendix A.1) and give the finite algebra needed here.

Let

g_{1} = X Z Z X I, g_{2} = I X Z Z X, g_{3} = X I X Z Z, g_{4} = Z X I X Z .

Tensor positions are $1, \dots, 5$ . Define

P = \frac{1}{16} j = 1 \prod^{4} (I + g_{j}), X = X X X X X, Z = Z Z Z Z Z .

The generators commute. The following table gives every nonidentity stabilizer product, with its overall sign omitted only because this table is used to inspect support.

Product	Unsigned word	Product	Unsigned word	Product	Unsigned word
$1$	$X Z Z X I$	$2$	$I X Z Z X$	$3$	$X I X Z Z$
$4$	$Z X I X Z$	$12$	$X Y I Y X$	$13$	$I Z Y Y Z$
$14$	$Y Y Z I Z$	$23$	$X X Y I Y$	$24$	$Z I Z Y Y$
$34$	$Y X X Y I$	$123$	$I Y X X Y$	$124$	$Y Z I Z Y$
$134$	$Z Y Y Z I$	$234$	$Y I Y X X$	$1234$	$Z Z X I X$

All fifteen words have weight four. Their distinct supports and words also show that the sixteen stabilizer products are distinct and that $- I$ is absent. Thus $P$ is an orthogonal projector of trace $32/16 = 2$ .

No nonidentity stabilizer product is diagonal in the computational basis. Hence $⟨ 00000 ∣ P ∣ 00000 ⟩ = 1/16$ . Fix the encoding explicitly by

∣ 0_{L} ⟩ = 4 P ∣ 00000 ⟩, ∣ 1_{L} ⟩ = X ∣ 0_{L} ⟩, V = ∣ 0_{L} ⟩ ⟨ 0 ∣ + ∣ 1_{L} ⟩ ⟨ 1 ∣ .

These vectors are normalized, have opposite $Z$ eigenvalues, and obey

X V = V X, Z V = V Z .

In this benchmark $b$ is trivial.

Displayed formulas in this passage

En02-B@1#eq:stabilizers · line 552
$g_{1} = X Z Z X I, g_{2} = I X Z Z X, g_{3} = X I X Z Z, g_{4} = Z X I X Z .$
En02-B@1#eq:encoding · line 597
$∣ 0_{L} ⟩ = 4 P ∣ 00000 ⟩, ∣ 1_{L} ⟩ = X ∣ 0_{L} ⟩, V = ∣ 0_{L} ⟩ ⟨ 0 ∣ + ∣ 1_{L} ⟩ ⟨ 1 ∣ .$
unlabeled · line 561
$P = \frac{1}{16} j = 1 \prod^{4} (I + g_{j}), X = X X X X X, Z = Z Z Z Z Z .$
unlabeled · line 607
$X V = V X, Z V = V Z .$