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

An explicit batch subsystem decoder

An explicit batch subsystem decoder · lines 764–811

En02-B@1#eq:reps, En02-B@1#eq:U

Set $A = {1, 2, 3}$ and $E = {4, 5}$ . Multiplication by stabilizers gives supported logical representatives

X_{A} = - Z_{1} X_{2} Z_{3}, Z_{A} = - Y_{1} Z_{2} Y_{3}, Y_{A} = - X_{1} Y_{2} X_{3} .

For example,

X g_{2} g_{3} g_{4} = - Z_{1} X_{2} Z_{3}, Z g_{3} = - Y_{1} Z_{2} Y_{3} .

Thus $X_{A} V = V X$ and $Z_{A} V = V Z$ .

Define the following unitary, with the rightmost gate applied first:

U = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3} .

Its chronological gate list is:

There are nine elementary gates and seven gate rounds.

Displayed formulas in this passage

En02-B@1#eq:reps · line 769
$X_{A} = - Z_{1} X_{2} Z_{3}, Z_{A} = - Y_{1} Z_{2} Y_{3}, Y_{A} = - X_{1} Y_{2} X_{3} .$
En02-B@1#eq:U · line 784
$U = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3} .$
unlabeled · line 776
$X g_{2} g_{3} g_{4} = - Z_{1} X_{2} Z_{3}, Z g_{3} = - Y_{1} Z_{2} Y_{3} .$