Manuscript Oai280-B@1 · Oai280-B.tex · proposition · 6d92ab75

Proposition: Exact code identification

Conventions and the explicit five-qubit encoding › The stabilizer convention · lines 314–362 · proof included or linked

Oai280-B@1#prop:stabilizers

The range of $V$ is the simultaneous $+ 1$ eigenspace of $g_{1}, \dots, g_{4}$ . Moreover,

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

Proof. Let

∣ G ⟩ = U_{G} ∣ + ⟩^{\otimes 5}, K_{j} = X_{j} Z_{j - 1} Z_{j + 1},

with cyclic indices. Conjugating $X_{j}$ by $U_{G}$ gives $K_{j}$ , so the $K_{j}$ stabilize $∣ G ⟩$ . They are independent: a product of a nonempty subset has a nonzero binary $X$ -support on precisely that subset.

Direct multiplication gives

g_{1} = K_{1} K_{4}, g_{2} = K_{2} K_{5}, g_{3} = K_{1} K_{3}, g_{4} = K_{2} K_{4} .

The four corresponding binary vectors are independent. Their common $+ 1$ eigenspace therefore has dimension $2$ ; equivalently the stabilizer projector has trace $32/16 = 2$ .

Writing $Z^{\otimes 5} = Z_{all}$ , the entrywise formula gives

∣ 0_{L} ⟩ = \frac{∣ G ⟩ + Z _{all} ∣ G ⟩}{2}, ∣ 1_{L} ⟩ = - \frac{∣ G ⟩ - Z _{all} ∣ G ⟩}{2} .

Every $g_{j}$ commutes with $Z_{all}$ , and hence stabilizes both columns. Thus their span is the whole two-dimensional stabilizer space.

The parity rule gives $Z V = V Z$ . Complementing every bit preserves parity up to one and satisfies

q (1 - x) = q (x) + 1

modulo two. Comparing the two columns in (eq:Vmatrix) therefore gives $X V = V X$ .

Displayed formulas in this passage

unlabeled · line 318
$X V = V X, Z V = V Z .$
unlabeled · line 325
$∣ G ⟩ = U_{G} ∣ + ⟩^{\otimes 5}, K_{j} = X_{j} Z_{j - 1} Z_{j + 1},$
unlabeled · line 335
$g_{1} = K_{1} K_{4}, g_{2} = K_{2} K_{5}, g_{3} = K_{1} K_{3}, g_{4} = K_{2} K_{4} .$
unlabeled · line 346
$∣ 0_{L} ⟩ = \frac{∣ G ⟩ + Z _{all} ∣ G ⟩}{2}, ∣ 1_{L} ⟩ = - \frac{∣ G ⟩ - Z _{all} ∣ G ⟩}{2} .$
unlabeled · line 357
$q (1 - x) = q (x) + 1$

Nearby in this section

section: The stabilizer convention