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

A sharp streaming and communication benchmark

Resource-restricted programmes › A sharp streaming and communication benchmark · lines 1030–1066

Oai280-C@1#eq:triplePaulis, Oai280-C@1#eq:R

We now restrict access to the regression triple $B = {1, 2, 3}$ of the five-qubit encoding. This is a precisely scoped resource theorem, not a claim about every access set of the connected encoding.

Multiplying the full logical Paulis by stabilizers gives

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

They satisfy $X_{B} V = V X$ and $Z_{B} V = V Z$ . Consequently

U_{B}^{†} X_{B} U_{B} = X \otimes I_{4}, U_{B}^{†} Z_{B} U_{B} = Z \otimes I_{4} .

Set

P_{12} = Z_{1} X_{2}, Q_{12} = Y_{1} Z_{2} .

These are commuting independent Pauli operators. Their common eigenprojectors are

Π_{pq} = \frac{1}{4} (I + p P_{12}) (I + q Q_{12}), p, q \in {+ 1, - 1} .

Each projector has rank one. Choose a one-qubit unitary $R_{pq}$ by

R_{pq}^{†} X R_{pq} = - pZ, R_{pq}^{†} Z R_{pq} = - q Y .

These two anticommuting axes specify a unitary up to global phase.

Displayed formulas in this passage

Oai280-C@1#eq:triplePaulis · line 1037
$X_{B} = - Z_{1} X_{2} Z_{3}, Z_{B} = - Y_{1} Z_{2} Y_{3} .$
Oai280-C@1#eq:R · line 1060
$R_{pq}^{†} X R_{pq} = - pZ, R_{pq}^{†} Z R_{pq} = - q Y .$
unlabeled · line 1044
$U_{B}^{†} X_{B} U_{B} = X \otimes I_{4}, U_{B}^{†} Z_{B} U_{B} = Z \otimes I_{4} .$
unlabeled · line 1050
$P_{12} = Z_{1} X_{2}, Q_{12} = Y_{1} Z_{2} .$
unlabeled · line 1055
$Π_{pq} = \frac{1}{4} (I + p P_{12}) (I + q Q_{12}), p, q \in {+ 1, - 1} .$

A sharp streaming and communication benchmark

Displayed formulas in this passage

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