Manuscript En02-D@4 · En02-D-v4.tex · lemma · e19ec871

Lemma: Signed logical representatives

The Clifford benchmark and its inherited unlocking state · lines 331–361 · proof included or linked

En02-D@4#lem:logical

For $ρ = (I + x X + y Y + z Z) /2$ ,

A_{L} (ρ) = \frac{1}{8} (I + x P_{X}^{(L)} + y P_{Y}^{(L)} + z P_{Z}^{(L)}),

where

L even L odd P_{X}^{(L)} - Z_{1} X_{2} Z_{3} - X_{2} Z_{3} P_{Y}^{(L)} - X_{1} Y_{2} X_{3} + Y_{1} X_{2} X_{3} P_{Z}^{(L)} - Y_{1} Z_{2} Y_{3} - Y_{1} Y_{3} .

The linear extension replaces $1, x, y, z$ by $tr A, tr (A X), tr (A Y), tr (A Z)$ .

Proof. Gate conjugation gives

U^{†} X_{2} U = - Z_{1} X_{2} Z_{3}, U^{†} Z_{2} U = - Y_{1} Z_{2} Y_{3} .

The identity $Y = i X Z$ gives the signed $Y$ word. Conjugating by $F$ , using $F X_{1} F = X_{1} Z_{2}$ , $F X_{2} F = Z_{1} X_{2}$ , and $F Z_{i} F = Z_{i}$ , gives the odd row. The displayed channel normal form proves the expansion.

Displayed formulas in this passage

unlabeled · line 334
$A_{L} (ρ) = \frac{1}{8} (I + x P_{X}^{(L)} + y P_{Y}^{(L)} + z P_{Z}^{(L)}),$
unlabeled · line 339
$L even L odd P_{X}^{(L)} - Z_{1} X_{2} Z_{3} - X_{2} Z_{3} P_{Y}^{(L)} - X_{1} Y_{2} X_{3} + Y_{1} X_{2} X_{3} P_{Z}^{(L)} - Y_{1} Z_{2} Y_{3} - Y_{1} Y_{3} .$
unlabeled · line 352
$U^{†} X_{2} U = - Z_{1} X_{2} Z_{3}, U^{†} Z_{2} U = - Y_{1} Z_{2} Y_{3} .$

Lemma: Signed logical representatives

Displayed formulas in this passage

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