Manuscript Oai280-C@1 · Oai280-C.tex · section · 7d44b8c0

A fully specified five-qubit tensor

A fully specified five-qubit tensor · lines 447–474

Oai280-C@1#eq:encoder

The five-qubit code is a standard exact quantum error-correcting code LaflammeEtAl. We specify its phases and normalization rather than importing a decoder convention.

Number physical qubits $1, \dots, 5$ and set

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

Tensor-product symbols are omitted in Pauli strings. Let $G = ⟨ g_{1}, g_{2}, g_{3}, g_{4} ⟩$ and

P = \frac{1}{16} g \in G \sum g, X = X X X X X, Z = Z Z Z Z Z .

Define

∣ 0 ⟩ = 4 P ∣ 00000 ⟩, ∣ 1 ⟩ = X ∣ 0 ⟩, V ∣ b ⟩ = ∣ b ⟩ .

Displayed formulas in this passage

unlabeled · line 454
$g_{1} g_{3} = X Z Z X I, = X I X Z Z, g_{2} g_{4} = I X Z Z X, = Z X I X Z .$
Oai280-C@1#eq:encoder · line 468
$∣ 0 ⟩ = 4 P ∣ 00000 ⟩, ∣ 1 ⟩ = X ∣ 0 ⟩, V ∣ b ⟩ = ∣ b ⟩ .$
unlabeled · line 462
$P = \frac{1}{16} g \in G \sum g, X = X X X X X, Z = Z Z Z Z Z .$

A fully specified five-qubit tensor

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