Manuscript En04-A@1 · En04-A.tex · theorem · bab77211

Theorem: Clifford constancy criterion

A genuine Clifford criterion and a responding deformation › Pauli compression for a Clifford encoder · lines 1031–1064 · proof included or linked

En04-A@1#thm:cliffordcriterion

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For each Hermitian residual Pauli $P_{B}$ , write

U^{†} (I_{R} \otimes P_{B}) U = s P_{C} \otimes P_{A}, s \in {- 1, 1} .

The complete complement is constant if and only if every such Pauli satisfies one of:

$P_{A}$ has an $X$ or $Y$ factor, so $⟨ 0^{m} ∣ P_{A} ∣ 0^{m} ⟩ = 0$ ;
$P_{C} = I_{C}$ .

Proof. Compression gives

V^{†} (I_{R} \otimes P_{B}) V = s ⟨ 0^{m} ∣ P_{A} ∣ 0^{m} ⟩ P_{C} .

The ancilla expectation is zero when an $X$ or $Y$ factor is present, and otherwise is one with the phase carried by $s$ . Thus every compression is scalar precisely under the stated condition.

Residual Paulis form an operator basis. The criterion for all of them is therefore equivalent to the observable criterion in Theorem thm:exact.

Displayed formulas in this passage

unlabeled · line 1034
$U^{†} (I_{R} \otimes P_{B}) U = s P_{C} \otimes P_{A}, s \in {- 1, 1} .$
unlabeled · line 1051
$V^{†} (I_{R} \otimes P_{B}) V = s ⟨ 0^{m} ∣ P_{A} ∣ 0^{m} ⟩ P_{C} .$

Theorem: Clifford constancy criterion

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